Cook the egg noodles according to package instructions. Drain and set aside.
In a large skillet, heat the vegetable oil over medium heat. Add the diced onion and sauté for about 3 minutes until translucent.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Increase the heat to medium-high and add the ground beef. Cook until browned, breaking it apart with a spatula, about 5-7 minutes.
Stir in the sliced bell pepper and broccoli florets. Cook for another 3-4 minutes until the vegetables are tender.
In a small bowl, whisk together the soy sauce, brown sugar, cornstarch, beef broth, ground ginger, and black pepper until smooth.
Pour the sauce over the beef and vegetable mixture in the skillet. Stir well to combine and cook for an additional 2-3 minutes until the sauce thickens.
Add the cooked egg noodles to the skillet and toss everything together until the noodles are well coated with the sauce.
Serve hot, garnished with sliced green onions and sesame seeds if desired.