In a large pot, combine beef brisket, water, crushed garlic, sliced ginger, and green onion. Bring to a boil, then reduce heat and simmer for 1.5 hours until the beef is tender.
Remove beef brisket and green onion from the broth. Strain the broth to remove solids. Let the broth cool completely, then season with soy sauce, sugar, and salt. Chill the broth in the refrigerator for at least 2 hours.
While the broth chills, slice the cooked beef brisket thinly. Thinly slice Korean radish and cucumber. Prepare boiled eggs and halve them. If using, thinly slice Asian pear.
Bring a large pot of water to a rolling boil. Add Korean buckwheat noodles and cook according to package instructions, typically 2-3 minutes.
Immediately drain the cooked noodles and rinse thoroughly under cold running water until completely chilled. Drain well.
Divide the chilled noodles into serving bowls. Pour the cold seasoned beef broth over the noodles.
Arrange the sliced beef, pickled radish, cucumber, and halved boiled eggs on top of the noodles. If desired, add sliced Asian pear. Sprinkle with roasted sesame seeds. Serve immediately with white vinegar and Korean mustard sauce on the side for individual seasoning.
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