Melt the butter in a large skillet over medium-high heat. Add the sliced onions, one Tablespoon of Worcestershire sauce, and ¼ cup of the beef broth to the skillet and stir to coat the onions.
Continue cooking the onions, stirring occasionally for 6-7 minutes until soft and golden.
Meanwhile, in a large bowl add the ground beef, egg, 1 Tablespoon of Worcestershire sauce, breadcrumbs, Italian seasoning, garlic powder, salt and pepper.
Use your hands or a large spoon to gently mix the beef mixture until everything is well combined.
Form the beef mixture into 5 even balls, then flatten into patties, set aside on a plate or cutting board.
Once the onions have softened, scoot them to the side and add the patties to the pan.
Fry the patties for 4 minutes per side, then, remove to a plate.
Stir the onions around in the pan, then sprinkle the flour over the onions and cook, stirring constantly for one minute.
Slowly pour the remaining beef broth into the skillet, stirring constantly and scraping any browned bits from the bottom of the pan.
Reduce the heat to medium-low, and add the patties back to the pan.
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.