Combine ground beef, bread crumbs, chopped onion, egg, salt, and black pepper in a bowl until evenly mixed. Shape beef mixture into 4 patties, about 3/4 inch thick.
Fry patties in a large skillet over medium heat until browned on both sides, about 10 minutes. Add beef broth, onion, and mushrooms; bring to a boil. Reduce heat to low, cover, and simmer until patties are no longer pink in the center, about 10 minutes more. Transfer patties to a platter and keep warm.
Bring onion mixture to a boil. Mix cornstarch and water in a small bowl; stir into onion mixture. Cook and stir until onion gravy is thickened, about 1 minute. Pour over patties to serve.
