Make the sauce by mixing soy sauce, water, brown sugar, hoisin, sesame oil, cornstarch, and red pepper flakes if using.
Cook the beef by heating oil in a pan, adding ground beef with salt and pepper, and cooking until browned. Drain extra fat.
Stir in garlic and ginger, cooking for 30 seconds.
Add carrots or bell peppers if using, and cook for 1–2 minutes.
Pour in the sauce and cook until it thickens, about 2 minutes.
Return the beef to the pan and mix well.
Stir in green onions, then serve over rice or noodles.