Agree with /u/X28.

Andrea's book should be considered as a primary text for Vietnamese cooking (much like David Thompson's Thai Food for Thai, or Paul Bocuse: The Complete Recipes for French).

Luke's books are great (as well as his shows that sort of accompany the books, or the other way around).

Asia's a big, ancient place. Even within each nation there are unique styles of regional and ethnic fare.

With that in mind, I'd love to see some recommendations here for awesome Indian, Filipino, Hmong, Uzbek, etc. cookbooks.

Japanese

Lets get beyond sushi and hibatchi.

Shizuo Tsuji's Japanese Cooking: A Simple Art is a great starting point. If you want to get technical you should check out Ando's Washoku or Hachisu's Preserving the Japanese Way.

If you want to start simple, Hachisu also has a great book on Japanese Farm Food. Ono and Salat have written a great noodle slurping opus in Japanese Soul Cooking.

Chinese

What we've come to think of as Chinese food in the US is a natural part of human appropriation of food styles, but with all due respect to Trader Vic's, crab rangoon and other buffet staples really aren't the real deal. Food in China is extremely regional. You don't have to go very deep to see the vast differentiation in spicy Schezwan recipes and Cantonese Dim Sum culture.

For your reading pleasure:

Mastering the Art of Chinese Cooking Eileen Yin-Fei Lo.

Breath of the Wok by Grace Young and Alan Richardson.

Phoenix Claws and Jade Trees by Kian Lam Kho and Jody Horton.

All Under Heaven: Recipes from the 35 Cuisines of China by Carolyn Phillips.

Some people might freak out that I'm placing Erway's The Food of Taiwan under the Chinese category, but I'm not going to get into a political debate here. Taiwan has had a lot of different culinary influences due to migration / occupation and that is really the take away here.

Go forth, make bao.

Korean

Korea is having it's moment right now and if you want the classics, Hi Soo Shin Hepinstall's Growing up in a Korean Kitchen is a good baseline. It has all the greatest hits.

You also can't cook Korean food without kimchi. The only book I've read is Lauryn Chun's The Kimchi Cookbook which is kind of underwhelming considering the hundreds of styles of Kimchi that have been documented. The process of making kimchi (kimjang) even has a UNESCO world heritage designation. With that in mind, I think it's only a matter of time before we see a English book on the subject that has depth.

Given the cuisine's popularity, there are several other cookbooks on Korean food that have recently been published within the last year or so, I just haven't gotten around to reading them yet, so I won't recommend them here.

Thai

David Thompson's Thai Food and Thai Street Food are both excellent. /u/Empath1999 's recommendation of Andy Ricker's Pok Pok is excellent but it focuses on Northern Thai cuisine, so if you want to venture into central and southern Thai fare, Thompson's the other farang of note.

Vietnamese

Nguyen's Into the Vietnamese Kitchen provides a nice survey to Vietnamese cooking. Charles Phan also has a couple of cookbooks that are quite good but I'm sure that there are zealots out there who would bemoan authenticity in either Vietnamese Home Cooking or The Slanted Door, but seriously, who gives a shit, the dude has Beard Awards under his belt for fuck's sake.

TL;DR OP means well but its long past time to bury "Asian" as a catch-all for such a large and diverse part of a continent, no?

FYI I'd recommend this book if you're into Thai Food

I got my first copy many years ago and it's a bit of a Thai food Bible.

Fuschia Dunlop is a good source for Chinese food. Her published recipe for Kung Pao Chicken is pretty killer. Eileen Yin-Fi Lo is also a well respected Chinese recipe author, check out My Grandmother's Chinese Kitchen.

For Thai Food, Andy Ricker's Pok Pok is pretty interesting (and the restaurants are pretty awesome). There's also a tome, simply called Thai Food from David Thompson, as an outsider, looks complete and exhaustive (it's also daunting to me, but nice to have).

Hot Sour Salty Sweet also features Thai (as well as other SE Asian flavors). And I really like Asian Flavors of Jean-Georges as a more upscale cookbook.

Also, I find this little, unsung book to be a great resource. It has fairly simple recipes that can yield some nice flavors, great for weeknight dishes.

And, Momofuku is a fun contemporary twist with some good basics, but it's not a beginner book by any stretch!

Finally, The Slanted Door is on my wishlist. Looks divine.

Well, I'm half-Chinese. I'll give you two cookbook recommendations which are full of recipes which really resonate with that part of my background:

• Mastering the Art of Chinese Cooking by Eileen Yin-Fei Lo. While I'm generally not big on Chinese cookbooks not specializing in one or two regional cuisines, this book gets a pass because it's so organized and pedantic. It builds itself up from simple to complex and includes recipes which build on each other. It also features a large section on ingredients. An additional pro is that it includes the Chinese characters which makes it easier to find the proper product at your Asian grocer. I love it so much that I lugged this book to Taiwan with me and used it as my cooking guide/reference.
  
• Land of Plenty by Fuchsia Dunlop. Of all the regional Chinese cuisines with which I have experience, I love the multi-layered flavors of Sichuan the most. It was through Dunlop's book that I first discovered this magnificent cuisine and it encouraged me to discover some of the Sichuanese restaurants in the Bay Area. Instructions are clear and she does a great job bringing Sichanese food to life. An absolute must own if you are at all interested in regional Chinese food. Her book on Hunanese food is also pretty killer.
  
  In addition to the aforementioned Chinese food, I'm just a fat piggy who loves to eat. Here are a few more recs:
  
  
• Thai Food by David Thompson. This is the bible of Thai food for English speakers. It's nearly 700 pages long and not a page is wasted on fluff. It's more than just a cookbook, it's a anthropological study on Thai people, their history and the way they eat. An immense book. If you are more into pictures, check out his book on Thai Street Food.
  
• Into the Vietnamese Kitchen by Andrea Nguyen. This was the book that really unlocked Vietnamese food for me. I adore how many fresh herbs/veggies are used and how it creates a complex, yet light, cuisine. And don't get me wrong when I say light...it's as full flavored as can be, but without heaviness. In the interim since this book came out, others have showed up on the market which are as good (see Charles Phan's recent book), but Nguyen's book will always have a special place in my heart.
  
• ad hoc at home by Thomas Keller. Thomas Keller is arguably the most important American chef of the past 20 years, so when he turns his sights on homestyle food, you can be sure it's done with correct technique and style. While this book isn't as notable as TFL cookbook or his sorta primer on sous vide cooking, I'm including it because it has recipes which people will actually use. Unparallelled technique, good recipes and delicious food equal a winning cookbook. One note: it's not dumbed down and some of the recipes take time, but everything I've ever made from it has been great.
  
• Alinea by Grant Achatz. {Disclaimer: I worked for Grant Achatz for a couple of years.} Everyone should own at least one cookbook which is completely out of reach, but serves to inspire. When you flip through this book, your jaw will drop and you will wonder, multiple times, "WTF?!?!?!" It's an amazing testament to how open and possible American cuisine is at the moment and you'll do yourself well to flip through it. Additionally, the photographs and the book itself are phenomenal. The paper, in particular, is well worth the price of admission. It's sexy shit, yo.
  
  Feel free to drop me a line if you need more recommendations. I've got quite the cookbook collection (I love to cook, it's not just cookbook porn) and love to share my thoughts.

Pick a classic in a cuisine with which you're generally unfamiliar but for which you feel confidant you can get good ingredients. A few ideas:

• Diana Kennedy - Essential Cuisines of Mexico or The Art of Mexican Cooking
  
  
• Barbara Tropp - The Modern Art of Chinese Cooking
  
  
• David Thompson - Thai Food
  
  
• Claudia Roden - The New Book of Middle Eastern Food
  
  
• Shizuo Tsuji - Japanese Cooking: A Simple Art
  
  You'll be forced to learn new techniques and deal with new ingredients, and get a sense of an entire cooking tradition. Any of those books will give you at least a small sense of the culture that inspired the cuisine, the human context, in addition to culinary knowledge.

Thai

http://www.amazon.com/Thai-Food-David-Thompson/dp/1580084621

http://www.amazon.com/Simple-Thai-Food-Classic-Recipes/dp/1607745232

http://www.amazon.com/Pok-Stories-Roadside-Restaurants-Thailand/dp/1607742888

My favourites for a few of the countries you list:

> Thailand

https://www.amazon.com/Thai-Food-David-Thompson/dp/1580084621/

> India

https://www.amazon.com/India-Cookbook-Pushpesh-Pant/dp/0714859028/

> Korea

https://www.amazon.com/Growing-up-Korean-Kitchen-Cookbook/dp/1580082815

a giant granite mortar and pestle is a good tool to have. This is a good book, as long as you can track down the ingredients. Andy Ricker's is probably also good, as I'm sure David Thompson's other book is too.

Yeah, yummo, love Tom Kha Gai, so quick and easy to make for something that impresses the pants off people. I usually add baby corn, bean sprouts, coriander to mine. I also add quite a bit more chilli.

Have you made Tom Yum? Similar but a thinner broth with no coconut milk made by boiling up prawn (shrimp) shells and heads.

David Thompson's book is amazing.

Basically gonna echo most of the answers already posted, but just to pile on:

• 8" chef's knife. 10" is longer than may be comfortable and 12" is longer than necessary, but 7" may start to feel a little short if she's ever slicing large melon or squash. I'm a casual knife nerd and I have knives by Wusthof, Victorinox, Shun and Mac. My favorite.
  
  
• This Dutch oven. Enameled and cast iron just like the Le Creuset that a few other comments have mentioned, but much, much cheaper. I own two and they're both great. I also have the non-enameled version for baking bread, but I don't recommend it for general use unless you're a Boy Scout. Here's an entertaingly-written blog post comparing the Lodge vs. Le Creuset in a short rib cookoff.
  
  
• This cutting board and this cutting board conditioner. The importance of an easy and pleasant to use prep surface can't be overstated. I'm listing this third on purpose; this is one of the most important things your kitchen can have. A recipe that calls for a lot of chopping is no fun when you're fighting for counter space to do the chopping, or doing it on a shitty plastic board.
  
  
• A cheap scale and a cheap thermometer. Seriously, these are as important as the cutting board.
  
  
• Just gonna crib this one right off /u/Pobe420 and say cheapo 8–10" (I recommend 10–12" but that's my preference) nonstick skillet. One note I'd add is that pans with oven-safe handles are a bit more dual-purpose than pans with plastic or rubberized handles. You can't finish a pork chop in the oven in a skillet with a rubberized handle. But one could say you shouldn't be cooking a pork chop on a nonstick pan to begin with. The important thing is to keep this one cheap: you're going to be replacing it every couple of years, there's no getting around that. For my money $30 or less, and $30 is pretty expensive for these things.
  
  
  

• Cookbooks
  

  Nothing inspires cooking like a good cookbook collection. The great news about cookbooks is that they're often bought as gifts or souvenirs and they make their way onto the used market cheap and in great condition. Here are my suggestions for a great starter shelf:
  
  

1. The Food Lab by J. Kenji López-Alt. I kind of hate that this is my number one recommendation, but I don't know your wife and I do know J. Kenji López-Alt. This one is brand new so you're unlikely to find it used and cheap, but as a catch-all recommendation it has to take first place. Moving on to the cheap stuff:
   
   
2. Regional French Cooking by Paul Bocuse. This is possibly the friendliest authoritative book on French food out there, and a hell of a lot easier to just dive into than Julia Child (Julia is the expert, and her book is an encyclopedia). Bocuse is the undisputed king of nouvelle cuisine and people like Eric Ripert and Anthony Bourdain (so maybe a generation ahead of you and I) came from him. Paul Bocuse is French food as we know it, and yet this book—an approachable, coffee-table sized thing—still has a recipe for fucking mac and cheese. It's outstanding.
   
   
3. Theory & Practice / The New James Beard by James Beard. These will completely cover your entire library of American cooking. Nothing else needed until you get region-specific. When you do, go for something like this.
   
   
4. Essentials of Classic Italian Cooking by Marcella Hazan. When she died, the NYT ran a second obituary that was just her recipe for bolognese.
   
   
5. Christ, top five. Who gets 5th? I'm going with From Curries To Kebabs by Madhur Jaffrey. Don't get bamboozled into buying "Madhur Jaffrey's Curry Bible" which is the same book, repackaged and priced higher. You want the one with the hot pink dust jacket, it's unmistakeable. This is one of those end-all books that you could cook out of for the rest of your life. It covers almost every diet and almost every country that Beard and Bocuse don't.
   
   
6. Honorable mentions: Here come the downvotes. Pok Pok by Andy Ricker. If you're American and you want to cook Thai, this is the one. Ten Speed Press can go home now. The Book of Jewish Food by Claudia Rosen (so close to making the list). I shouldn't need to say much about this; it's the book of diasporic Jewish food, which means it covers a lot of time and almost every possible country. It's a no-brainer. Thai Food by David Thompson (a perfect oral history of Thai food for English speakers, only it doesn't include Pok Pok's precise measurements, which in practice I've found important). Flour Water Salt Yeast by Ken Forkish. Not for someone who just wants to become a baker, this book is for someone who wants to make Ken Forkish's bread. And for a casual bread baker I can't imagine a better introduction. Pleasures of the Vietnamese Table by Mai Pham. Andrea Nguyen is out there and Andrea Nguyen is awesome, but I really like Mai Pham's book. It's accessible, reliable and regional. You don't get the dissertation-level breakdown on the origins of chicken pho that you get from Andrea, but the recipe's there, among many others, and it's fucking outstanding. Veganomicon by Isa Chandra Moskowitz and Terry Hope Romero. This vegan cookbook is dope as hell and will really expand your imagination when it comes to vegetables. This could actually have been number five.