In a small bowl, whisk together soy sauce, brown sugar, cornstarch, and sesame oil.
In a large pan, heat vegetable oil over medium-high heat.
Add garlic and ginger to the pan and cook until fragrant, about 2 minutes.
Add the chicken and steak to the pan and cook until browned, about 5-7 minutes.
Add the broccoli and carrot to the pan and cook until tender.
Stir in the sauce mixture and cook until thickened.
Meanwhile, cook instant ramen according to package instructions.
Serve the Mongolian beef over the cooked ramen noodles.
Garnish with scallions, red pepper flakes, and toasted sesame seeds.
