Whisk broth, soy sauce, brown sugar, gochujang, vinegar, sesame oil, garlic, and ginger in the slow cooker.
Nestle the beef into the sauce. Scatter onion and carrots around it. Sprinkle red pepper flakes if using.
Cover and cook on Low 7–8 hours or High 3–4 hours, until beef is fork-tender.
Transfer beef to a board. Shred with two forks and discard any excess fat.
Stir cornstarch with water to make a smooth slurry. Whisk into the cooker liquid.
Return shredded beef to the sauce. Cover and cook 10–15 minutes until glossy and slightly thickened.
Boil noodles in salted water until just tender. Drain well.
Toss noodles with the beef and sauce, or plate noodles and spoon beef over top.
Finish with green onions and sesame seeds. Serve hot.