In a bowl, combine the sliced beef, soy sauce, and cornstarch. Mix well and let it marinate for about 15 minutes.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat. Add the marinated beef and stir-fry for about 3-4 minutes until browned. Remove the beef from the skillet and set aside.
In the same skillet, add the remaining tablespoon of oil. Add the sliced onion, bell pepper, and broccoli. Stir-fry for about 5 minutes until the vegetables are tender-crisp.
Add the minced garlic, ginger, and black pepper to the skillet. Stir for about 30 seconds until fragrant.
Return the cooked beef to the skillet and add the cooked udon noodles. Toss everything together until well combined and heated through, about 2-3 minutes.
Serve hot, garnished with chopped green onions.