Not all Korean BBQ is equal, the best is Han-U 한우, Korean domestic beef. You can only experience this special taste in Korea because it is not readily exported or available outside of Korea. Japan has Kobe beef, Korean has Han-U. Han-U is fatty and melts in your mouth with some chew without being overwhelmingly fatty like some cuts of Kobe beef. If you are ever in Korea this is something you must definitely try.
My favorite place to eat this changes from time to time, but currently it is Changgo43 창고43 with locations in Yeouido and Myeongdong. I like this particular restaurant for Han-U because the meat comes in larger slabs, so you get to really taste the flavor and texture moreso than with the smaller thinner pieces at other restaurants. Most restaurants grill over a charcoal pit but this restaurant grills the meat in a heavy cast iron skillet.
The menu in Korean is on Changgo43′s website here. The menu is very short with 3 types of meat, some soups/stews, and fried rice “bokkeumbap.” The three types of meat are the Changgo Special KRW 36,000 / $35, Tenderloin “Anshim” 36,000 / $35, Sirloin “Deungshim” KRW 51,000 / $50. Prices are for one order which is 150 grams or 5oz. Each of the larger pieces in the video is one order. (Assumes a $1=KRW 1,020 exchange rate.) I tried one order of each and the $50 Sirloin is by far the best; the waitress told us they sell the most of this one. The soups and stews are $5-$6 and fried rice is $3. The fried rice is cooked at the end of the meal in the residual beef fat with kimchi and scallions.
Han-U is very expensive in Korea as it is the rare top quality meat from only domestic cattle, so you will see a lot of people on corporate tabs in the restaurant after work. The smell from grilling permeates your clothes, so the restaurant provides large plastic bags for you to stuff your jackets into so at least your jacket does not smell like a Korean cow. They also provide a black apron to protect your clothing. You will need to shower after eating this, and some dogs on the street may try to follow you home. Despite the pervasive odor and high cost this is truly worth it.
Where to eat Han-U in Korea: My prior favorite place to eat Han-U before Changgo43 was Moodeungsan (무등산) in Cheongdamdong. This is a uber fancy two story bbq restaurant. As soon as you walk in you will see fancy looking people grilling meat and a wall of signatures from celebrities who have dined there like Lady Gaga and Britney Spears. All of their meat is really delicious and their nengmyun is especially good. I have also heard good things about Twobbul 투뿔등심 but have yet to try.