Heat oven to 400° F.
Place the beef in a large skillet and cook over medium-high heat until no trace of pink remains, about 5 minutes.
Spoon off and discard any fat. Stir in the ketchup and Worcestershire. Add the vegetables and cook, stirring, for 1 minute. Spoon into a baking dish.
Mix the cheese (if desired) with the potatoes in a bowl. Spread over the beef and bake until heated through, 10 minutes. Divide among plates.