01 - In a mixing bowl, combine ground beef, breadcrumbs, egg, Worcestershire sauce, garlic powder, onion powder, salt, and black pepper. Mix until just combined, then shape into 4 oval-shaped patties.
02 - Heat a large skillet over medium-high heat and add a light coating of oil if needed. Sear the patties for 3 to 4 minutes per side until well browned. Remove from the skillet and set aside; they do not need to be fully cooked at this stage.
03 - Reduce the heat to medium. In the same skillet, melt the butter and add the sliced onions. Cook, stirring frequently, until the onions are golden and deeply caramelized, about 10 to 12 minutes.
04 - Sprinkle the flour over the caramelized onions and cook for 1 to 2 minutes, stirring constantly to form a roux.
05 - Slowly pour in the beef broth while stirring continuously. Add the Worcestershire sauce and dried thyme, then bring the mixture to a simmer.
06 - Return the seared hamburger steaks to the skillet, spooning the onion gravy over them. Cover and simmer for 10 to 15 minutes until the steaks are cooked through and the gravy has thickened.
07 - While the steaks braise, brush the split sandwich rolls with melted butter and toast them in a skillet or under a broiler until golden brown.
08 - Place each cooked hamburger steak on the bottom half of a toasted bun. Spoon generous amounts of onion gravy over the top, add cheese if desired, and cap with the top half of the bun. Serve immediately while hot.