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