Place steak in a large resealable bag. Whisk together soy sauce, 1 tablespoon corn starch, brown sugar, garlic, ginger, red pepper flakes, black pepper, and sesame oil. Add sauce to the bag, press out any excess air, and seal. Chill for 10 minutes.
Drizzle a large pan or skillet over medium heat with the vegetable oil. Use tongs or a fork to transfer beef strips to the pan (do not discard marinade in the bag!). Add carrots, snow peas, and mushrooms to the pan. Saute for 4-6 minutes until browned.
Add marinade from the bag to the pan. Bring to a boil. Whisk remaining tablespoon of corn starch into the beef broth and stir into the pan until sauce thickens. Reduce to a low simmer.
Stir in cooked noodles, garnish with sesame seeds and green onions if desired and serve.