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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.
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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.
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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.
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Dissolve the cornstarch in the water in a small bowl. Stir into the mushroom mixture and bring to a boil. Cook, stirring, until thickened.
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Pour the mushroom gravy over the patties and serve.