In a large skillet, cook beef over medium-high heat, stirring often to break up lumps, until browned, 5 to 7 minutes. Drain off excess fat and rinse.
Stir in green pepper, onion, cucumber and tomato. Cook, stirring occasionally, until onion is tender, about 5 minutes. Stir in tomato sauce, ketchup, Worcestershire Sauce, salt and pepper to taste. Heat to boiling, reduce heat to low, and simmer 10 minutes.
Spoon mixture over muffin halves and serve open face.