Finely chop the onion.
Mix ground beef with onion, breadcrumbs, egg, milk, salt, and pepper thoroughly.
Shape the mixture into round patties, heat oil in a pan, and grill on both sides.
Once the beef is cooked, add onion and butter to the pan with the remaining oil and sauté.
Add tomato ketchup, Worcestershire sauce, and water, then bring to a boil to make the sauce.
Pour the sauce over the grilled Hamburg steak.