In a large bowl, mix together the ground beef, egg, onion soup mix, bread crumbs, Worcestershire sauce, and black pepper until well combined. Form into 4 balls and flatten into patties
Heat the oil in a large skillet over medium heat. Fry the patties and sliced onions in the skillet until the patties are nicely browned, about 4-5 minutes per side. Remove the beef patties to a plate and set aside
Add butter to hot skillet until melted. Sprinkle flour over the butter, onions, and drippings. Whisk in the flour scraping bits of beef off the bottom, whisk constantly until mixture slightly darkens in color. Slowly mix in the beef broth whisking constantly until well combined. Simmer and whisk over medium heat until gravy thickens about 5 minutes
Turn the heat to low and return beef patties to the gravy and simmer uncovered for another 10 minutes, serve