01 - In a large mixing bowl, combine the ground beef, breadcrumbs, egg, Worcestershire sauce, garlic powder, onion powder, salt, and black pepper. Mix gently until just incorporated, being careful not to overwork the meat. Shape the mixture into 4 even oval-shaped patties, about 3/4 inch thick.
02 - Heat a large skillet over medium-high heat and add a light coating of oil if the pan is dry. Sear the patties for 3 to 4 minutes per side until a deep golden-brown crust forms. Remove the steaks from the skillet and set them aside on a plate; they will finish cooking in the gravy.
03 - Reduce the skillet heat to medium. Add the butter and let it melt, then add the thinly sliced onions. Cook, stirring frequently, until the onions are deeply golden and caramelized, about 10 to 12 minutes. Scrape up any browned bits from the bottom of the pan for added flavor.
04 - Sprinkle the flour evenly 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 whisking steadily to prevent lumps. Stir in 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 onions. Spoon the gravy generously over each patty. Cover the skillet with a lid 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, coating 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. Ladle a generous amount of onion gravy over the top. Add sliced cheese if desired, then cap with the top half of the bun. Serve immediately while hot.