Combine the ground beef, bread crumbs, egg, milk, chopped onion, garlic, Worcestershire sauce, Dijon mustard, salt, and pepper in a large bowl. Mix gently to avoid tough patties.
Divide the mixture into 4 equal portions and form into oval patties about 1/2 inch thick.
Heat olive oil in a skillet over medium-high heat. Cook patties for 4-5 minutes per side until browned. Remove and set aside.
Prepare the Gravy: In the same skillet, add butter and sliced onions. Cook until soft and caramelized, about 8-10 minutes.
Sprinkle flour over the onions and cook for 1-2 minutes.
Gradually whisk in beef broth, then add Worcestershire sauce and Dijon mustard. Simmer until thickened.
Return patties to the gravy. Cover and simmer for 10 minutes until cooked through.
Taste the gravy and adjust with salt and pepper.
Serve hot over mashed potatoes, rice, or vegetables.