In a large skillet over medium high heat melt butter. Add onions slices, 1 TBS of worchestershire sauce and 1/4 cup beef broth. Stir occasionally for 6 minutes until softened and lightly browned.
While onions are browning: In a large bowl combine ground beef, egg, panko or bread crumbs, parmesan cheese, italian seasoning, garlic powder, salt & pepper. Form into 4-6 patties.
Remove onions from skillet once done and set aside. Place meat patties into pan and brown for 4 minutes on one side, turn patties over and brown an additional 4 minutes.
Remove patties from pan and add onions back in. Sprinkle onions with flour and cook for 1 minute while stirring. Add in rest of beef broth and french onion soup. Bring to a boil while stirring. Lower heat to a medium low and place meat patties back into skillet.
Cover meat patties with cheese slices or 1/4 cup of shredded cheese per patty, place lid on skillet and let simmer for 6-8 minutes until cheese is melted, patties are cooked through and gravy has thickened.