Bring a large pot of salted water to a boil. Add the udon noodles and cook according to the package instructions until al dente. Drain and set aside.
In a large skillet or wok over medium-high heat, add the lean ground beef. Cook until browned, breaking it apart with a spatula as it cooks, about 5-7 minutes. Drain any excess fat.
Once the beef is cooked, add the minced garlic and ginger to the skillet. Sauté for about 1-2 minutes until fragrant, stirring frequently to prevent burning.
In a medium bowl, whisk together the soy sauce, brown sugar, beef broth, hoisin sauce, red pepper flakes (if using), and cornstarch. This will create a flavorful sauce that will coat the beef and noodles.
Pour the sauce over the cooked ground beef in the skillet. Stir well to combine. Reduce heat to medium-low and let it simmer for about 3-5 minutes until the sauce thickens slightly.
Add the cooked udon noodles to the skillet, tossing them gently in the sauce until they are fully coated. If the mixture seems too thick, add a bit of water until you reach your desired consistency.
Remove from heat and serve the Mongolian Ground Beef and Noodles immediately. Garnish with chopped green onions or sesame seeds if desired.