In a large bowl, combine the ground beef, breadcrumbs, egg, milk, salt, and pepper. Mix gently until just combined — avoid overworking.

Shape the mixture into 4 to 6 even patties, about 3/4-inch thick.

Heat a tablespoon of oil or butter in a large skillet over medium heat. Brown the patties 3–4 minutes per side until a deep crust forms. They do not need to be fully cooked through. Remove the patties and set aside.

In the same skillet, add the thinly sliced onions. Sauté, stirring occasionally, until softened and beginning to brown, about 6–8 minutes. Scrape up any browned bits from the pan.

Sprinkle the flour over the onions and cook, stirring, for 1 minute to remove the raw flour taste.

Gradually pour in the beef broth while stirring to avoid lumps. Add the Worcestershire sauce and thyme. Bring the mixture to a simmer and let it thicken slightly.

Return the browned patties to the skillet, nestling them into the gravy. Cover and simmer gently for 12–15 minutes, or until the patties are cooked through (internal temperature 160°F/71°C) and the gravy has thickened to your liking. Adjust salt and pepper to taste.