In a bowl, combine the ground beef, bread crumbs, milk, egg, finely chopped onion, minced garlic, Worcestershire sauce, salt, and pepper. Mix well until just combined.

Divide the beef mixture into four equal portions and shape each into a patty about 1-inch thick.

In a large skillet, melt 1 tablespoon of butter over medium heat. Add the patties to the skillet, cooking for about 4-5 minutes on each side until they are browned and cooked through. Remove the patties from the skillet and set aside.

In the same skillet, add the remaining tablespoon of butter and the sliced mushrooms. Sauté until the mushrooms are browned, about 2-3 minutes.

Sprinkle the flour over the mushrooms, stirring to coat.

Gradually pour in the beef broth while stirring constantly to prevent lumps. Bring the mixture to a simmer, allowing it to thicken slightly.

Return the patties to the skillet, spooning the gravy over them. Let them simmer in the gravy for about 5 minutes, then serve hot.