Mix the beef, bread crumbs, onions, egg, Worcestershire sauce, salt, and pepper with your hands in a medium bowl. Shape into 4 patties, each about 3/4 to 1 inch thick.
Melt the butter with the oil in a large skillet over medium-high heat. Add the patties and fry for 4 to 5 minutes on each side. Use a spatula to transfer the patties from the skillet to an oven-proof plate; keep warm.
Add the sliced onion and mushrooms to the skillet and saute until the mushrooms give up their juice, about 8 minutes. Stir in the broth, using a wooden spoon to scrape up any browned bits from the bottom of the skillet. Heat to boiling, then reduce the heat, cover, and simmer for 10 minutes.
Dissolve the cornstarch in the water in a small bowl. Stir into the mushroom mixture and bring to a boil. Cook, stirring, until thickened.
Pour the mushroom gravy over the patties and serve.