Start by boiling water in a large pot. Add the egg noodles and cook according to the package instructions, usually about 5-7 minutes. Once cooked, drain and set aside.
In a large skillet or wok, heat the sesame oil over medium-high heat.
Add the chopped onion and sauté for 2-3 minutes until slightly translucent.
Incorporate the minced garlic and ginger into the skillet, cooking for an additional minute until fragrant.
Add the ground beef to the skillet, breaking it up with a spoon. Cook until browned and cooked through, about 5-7 minutes.
Stir in the soy sauce and brown sugar, allowing the flavors to combine for about 2 minutes.
Mix in the vegetables and cook until tender, about 3-5 minutes. Season with salt and pepper to taste.
Add the cooked noodles to the beef mixture, tossing everything together until well coated in the sauce. Heat for an additional minute if necessary.
Serve the Mongolian ground beef noodles hot, garnished with sliced green onions and sesame seeds if desired.