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 oval-shaped patties, pressing a small indent in the center of each to promote even cooking.
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 hamburger steaks 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. In the same skillet, melt the butter and add the thinly sliced onions. Cook, stirring frequently, until the onions are golden and fully caramelized, approximately 10 to 12 minutes. Scrape up any browned bits from the bottom of the pan to incorporate into the fond.
04 - Sprinkle the flour evenly over the caramelized onions and cook for 1 to 2 minutes, stirring constantly to form a roux and cook out the raw flour taste.
05 - Slowly pour in the beef broth while whisking continuously to prevent lumps. Stir in 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 generously over the top. Cover 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 in a skillet or under a broiler until golden and lightly crisp.
08 - Place a cooked hamburger steak on the bottom half of each toasted bun. Ladle a generous spoonful of onion gravy over the steak. Add sliced cheese if desired, then cap with the top half of the bun. Serve immediately while hot.