Cook the egg noodles according to package instructions. Drain and set aside.
In a large skillet or wok, heat the vegetable oil over medium-high heat. Add the sliced onion and sauté for 2-3 minutes until softened.
Add the minced garlic and cook for an additional 30 seconds until fragrant.
Increase the heat to high and add the ground beef. Cook until browned, breaking it apart with a spatula, about 5-7 minutes.
Stir in the bell pepper and broccoli, cooking for another 3-4 minutes until the vegetables are tender-crisp.
In a small bowl, whisk together the soy sauce, brown sugar, cornstarch, beef broth, ground ginger, and red pepper flakes (if using).
Pour the sauce over the beef and vegetables in the skillet. Stir well to combine and cook for 2-3 minutes until the sauce thickens.
Add the cooked egg noodles to the skillet, tossing everything together until well coated in the sauce. Cook for an additional 2 minutes to heat through.
Serve hot, garnished with chopped green onions and sesame seeds.