Brown the ground beef in a medium skillet over medium-high heat, until meat is no longer pink. Add onion and mushrooms to mixture, saute.
Drain excess fat and set aside.
Melt the butter or margarine in a medium saucepan over medium heat. Slowly add the flour, stirring constantly, to form a brown roux.
Pour in beef broth and milk, mixing well. Add ground beef, then season with salt and pepper to taste. Serve this dish over mashed potatoes, toast or english muffins.