Make the patties: Add the ground beef, panko, grated onion, egg, 1 tablespoon Worcestershire, Creole seasoning, and a generous pinch of black pepper to the bowl. Mix gently with your hands until just combined — overworking the meat makes the patties tough. Form into 4 oval patties, about 3/4-inch thick.
Brown the patties: Heat a tablespoon of oil in your skillet over medium-high heat. Brown the patties for 3-4 minutes per side, until a deep crust forms. Don't crowd the pan — work in batches if needed. Transfer the patties to a plate. (Photo tip: You're looking for a deep brown crust, not just gray — that's the flavor foundation of the whole gravy.)
Sauté the aromatics: Reduce the heat to medium. Add the sliced onions and mushrooms with a pinch of salt. Cook, stirring occasionally, for 6-8 minutes until softened and deeply browned. Scrape up the browned bits from the bottom of the pan as they cook.
Make the gravy base: Sprinkle the flour over the vegetables and stir constantly for 1 full minute. This cooks out the raw flour taste so your gravy doesn't taste like paste.
Build the gravy: Slowly pour in the beef broth while whisking constantly to prevent lumps. Add the remaining 1 tablespoon Worcestershire and a few sprigs of fresh thyme if you have them. Bring to a gentle simmer. (Photo tip: Whisk continuously at the edges to catch any stray lumps before they form.)
Simmer: Return the patties and any collected juices to the skillet. Reduce the heat to low, cover, and let it simmer for 20 minutes. The gravy will thicken and deepen in flavor. (Photo tip: The gravy should coat the back of a spoon — that's your signal it's ready.)
Finish and serve: Remove the thyme sprigs. Stir in the butter until melted for a glossy finish. Taste the gravy and adjust salt and pepper. Serve over mashed potatoes or buttered egg noodles.