Mix beef, breadcrumbs, egg, chopped onion, Worcestershire, salt, and pepper. Shape into 6 oval patties.
Heat 1 tbsp oil in a skillet and brown patties on both sides. Remove to a plate.
In the same skillet, cook sliced onion in remaining oil until soft and lightly browned (5–10 minutes).
Whisk broth, Worcestershire, flour (and ketchup if using). Add to onions and cook, stirring, until thickened.
Return patties and juices to the skillet, spoon gravy over, cover, and simmer about 10 minutes until cooked through.
