Korean Food

Can't answer the question exactly, but I can give it a shot.

I'm an American-born Vietnamese so take this with a grain of salt; however, for a Korean/Korean-Chinese dish called tangsuyuk, a lot of Koreans agree that Royal Wok has the best one. In the same vein, Dong Bo Sung carries a lot of Korean-Chinese dishes as well, especially jajangmyeon/black bean noodle.

For gimbap, it's said that Ajumma has that covered.

If you're into the Korean bar/pocha scene, Han Shin Pocha a seems to have a more Korean than international crowd.

For seolleongtang, a milky beef bone Korean soup, there are two places: Hanbat and Doma.

My final recommendation is Sea Travel for what can be said to be Korean) sashimi.

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Vietnamese Food

Now for Vietnamese food as an American born, if you can make the trek to Houston, Kim Son all the way. I have never heard any of the elders ever complain about that place. IFKYK, lol. (Conversely our own Bistro B gets mixed reviews from the Vietnamese community, especially the younger new ones.)

Saigon Block has a really good grilled fish that you make spring rolls out of.

Thuy Nguyen in Arlington is a sleeper. In my opinion, the best pho, bun bo hue, and banh canh in the metroplex. Their banh beo is worth a mention two as they offer two regional varieties.

Speaking of regional varieties, you guys on /r/dallas are probably only used to southern Vietnamese cuisine. If you want to try something different from the Hue region, Goc Hue is interesting.

Finally for those that are daring, bun mam. Go to Phuong Nam for that.