Preheat the broiler and line the bottom of a broiler pan with foil. Lightly mix the beef with 1 teaspoon salt and 1/2 teaspoon pepper in a large bowl. Gently form into 4 patties, about 3/4 inch thick. Arrange on the prepared pan and brush both sides with olive oil. Broil until browned, about 4 minutes per side for medium rare.
Meanwhile, heat 2 teaspoons olive oil in a large skillet over medium-high heat. Add the onion and cook, stirring, 2 minutes. Add the mushrooms and cook 2 more minutes. Add the Worcestershire sauce and cook until the vegetables are tender, about 2 more minutes. Season with salt and pepper and remove from the heat. Combine the ketchup and mustard in a small bowl and set aside.
Top each patty with one-quarter of the mushroom mixture and 1 slice cheese. Return to the broiler to melt the cheese, about 1 minute.
Spread the buns with the ketchup mixture; sandwich with the patties and some lettuce. Serve with chips and/or pickles.