01 - In a mixing bowl, combine ground beef, breadcrumbs, egg, Worcestershire sauce, garlic powder, onion powder, salt, and black pepper. Mix until just combined, being careful not to overwork the meat. Shape into 4 oval-shaped patties, about ¾ inch thick.
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 a deep brown crust forms. Remove from the skillet and set aside; they will finish cooking in the gravy.
03 - Reduce the heat to medium. Melt the butter in the same skillet and add the thinly 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 and cook off the raw flour taste.
05 - Slowly pour in the beef broth while stirring to prevent lumps. Add the Worcestershire sauce and dried thyme. Bring the mixture to a gentle 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 to a rich consistency.
07 - While the steaks braise, brush the split sandwich rolls with melted butter. Toast them in a separate skillet or under a broiler until golden brown and crisp.
08 - Place each cooked hamburger steak on the bottom half of a toasted bun. Spoon generous amounts of onion gravy over the top. Add sliced cheese if desired, then cap with the top half of the bun. Serve immediately while hot.