In a large skillet over medium heat, cook onions in 5 teaspoons oil until golden brown. Add salt and pepper. Cover and keep warm.
In a large bowl, combine Worcestershire sauce and ground beef, mixing well. Divide into 8 portions. Shape into 4-inch patties. Place cheese on each, and top with remaining patties. Press down gently, sealing the edges well. Grill patties 14 to 18 minutes. Turn halfway through. Place patties on toasted buns. Top with onion mixture, then place bun on top of onions. Makes 4 servings.