01 - In a mixing bowl, combine ground beef, breadcrumbs, egg, Worcestershire sauce, garlic powder, onion powder, salt, and black pepper. Mix gently until just combined, being careful not to overwork the meat. Shape into 4 uniform oval patties, approximately 3/4 inch thick.
02 - Heat a large skillet over medium-high heat and add a light coating of oil if the pan is not well-seasoned. Sear the patties for 3 to 4 minutes per side until a deep golden-brown crust forms. Remove from the skillet and set aside on a plate; they will finish cooking in the gravy.
03 - Reduce the heat to medium and melt the butter in the same skillet, scraping up any browned bits from the bottom. Add the thinly sliced onions and cook, stirring frequently, for 10 to 12 minutes until they are deeply golden and softened.
04 - Sprinkle the flour evenly over the caramelized onions and stir constantly for 1 to 2 minutes to cook out the raw flour taste and form a light roux.
05 - Slowly pour in the beef broth while stirring continuously to prevent lumps. Add the Worcestershire sauce and dried thyme. Bring the mixture to a gentle simmer, allowing it to begin thickening.
06 - Return the seared hamburger steaks to the skillet, nestling them into the onion gravy. Spoon gravy generously over each patty. Cover the skillet and simmer for 10 to 15 minutes until the steaks are cooked through to an internal temperature of 160°F and the gravy has thickened to a rich consistency.
07 - While the steaks braise, brush the cut sides of the sandwich rolls with melted butter. Toast them in a separate skillet over medium heat or under a broiler until golden and lightly crisped.
08 - Place each cooked hamburger steak on the bottom half of a toasted bun. Spoon a generous amount of onion gravy over the top. Add sliced cheese if desired, then cap with the top bun. Serve immediately while hot.