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Melt the butter in a large skillet over medium-high heat. Add the sliced onions, one Tablespoon of Worcestershire sauce, and 1/4 cup of the beef broth to the skillet and stir to coat the onions.
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Continue cooking the onions, stirring occasionally for 6-7 minutes until soft and golden.
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Meanwhile, in a large bowl add the ground beef, egg, 1 Tablespoon of Worcestershire sauce, breadcrumbs, Italian seasoning, garlic powder, salt and pepper.
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Use your hands or a large spoon to gently mix the beef mixture until everything is well combined.
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Form the beef mixture into 5 even balls, then flatten into patties, set aside on a plate or cutting board.
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Once the onions have softened, scoot them to the side and add the patties to the pan.
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Fry the patties for 4 minutes per side, then, remove to a plate.
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Stir the onions around in the pan, then sprinkle the flour over the onions and cook, stirring constantly for one minute.
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Slowly pour the remaining beef broth into the skillet, stirring constantly and scraping any browned bits from the bottom of the pan.
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Reduce the heat to medium-low, and add the patties back to the pan.
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Top each patty evenly with the cheese. Cover the skillet and simmer for 6-8 more minutes until cheese is melted, gravy is thickened, and the meat is cooked through.
