In a large skillet heat the 4 teaspoons oil over medium heat. Add onion; cook about 10 minutes or until golden, stirring occasionally. Stir in salt and the 1/4 teaspoon pepper. Cover; keep warm.
Meanwhile, in a large bowl combine beef, Worcestershire sauce, the 1/2 teaspoon pepper, and the garlic. Divide into eight equal portions. Shape each portion into a 4-inch-diameter patty. Place one-fourth of the cheese on each of four of the patties. Top with remaining patties, pressing down lightly and sealing edges well.
Place burgers on the grill rack directly over medium heat; grill for 14 to 18 minutes or until done (160 degrees F), turning once halfway through grilling.
Meanwhile, brush bread slices lightly with the 1 tablespoon olive oil. Add bread slices to grill rack for the last 2 to 3 minutes of grilling time or until toasted, turning once. Serve patties on toasted bread slices topped with onion mixture. Makes 4 servings.