In a bowl, mix 2 tbsp. soy sauce, 1 tbsp. each vinegar and sesame oil, 1/4 cup green onions, and the bread crumbs and ground beef. Finely chop 1/4 cup kimchi and add to beef mixture. With wet hands, form meatballs the size of ping-pong balls.
Heat vegetable oil in a large nonstick frying pan over medium-high heat. Add meatballs and cook, turning, until well browned and cooked through, about 12 minutes. Lift meatballs to a rimmed baking pan, cover with foil, and keep warm.
Meanwhile, combine remaining 1 tbsp. soy sauce, 2 tbsp. each vinegar and sesame oil, and 1/4 cup each green onions and kimchi juice. Stir in chili garlic sauce.
Cook noodles according to package instructions, drain, and return to warm pot. Toss noodles with sauce and remaining 1 cup kimchi.
Divide noodles among 4 bowls, top with meatballs, and sprinkle with remaining 1/4 cup green onions and the sesame seeds.