Combine ground beef, bread crumbs, chopped onion, egg, salt, and black pepper in a large bowl until evenly mixed. Shape beef mixture into 4 patties, about ¾-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.
