In a large bowl, combine ground beef, breadcrumbs, egg, grated onion, Worcestershire sauce, salt, and pepper until just incorporated.
Form the mixture into four equal oval-shaped patties approximately 3/4-inch thick.
Heat vegetable oil in a large skillet over medium-high heat and sear patties for 3 to 4 minutes per side until browned. Remove patties and set aside.
In the same skillet, melt butter and add sliced onions. Cook for 8 to 10 minutes until onions are soft and caramelized.
Sprinkle flour over the onions and stir constantly for 1 minute to cook out the raw flour taste.
Gradually pour in the beef broth while whisking to prevent lumps. Add dried thyme and bring to a simmer.
Return the patties to the skillet and spoon gravy over them. Reduce heat to low and simmer for 5 to 10 minutes until the sauce has thickened and patties are cooked through.
Serve warm, typically over mashed potatoes or egg noodles.