Heat a large skillet over medium heat. Add oil; swirl. Add onion; cook 7 minutes or until tender, stirring occasionally. Add beer; bring to a boil over medium-high heat. Reduce heat. Simmer 20 minutes or until onion is golden and liquid almost evaporates; stir occasionally. Cool. Preheat grill to medium-high heat. Remove 1/4 cup onion mixture from pan; finely chop. Combine chopped onion mixture and turkey. Divide mixture into 4 equal portions; gently shape each into a 1/2-inch-thick patty. Press a nickel-sized indentation in center of each patty. Sprinkle with salt and pepper. Place patties on grill rack coated with cooking spray; grill 5 minutes. Turn patties; grill 3 minutes. Top with cheese; grill 1 minute or until cheese melts and turkey is done. Combine mayonnaise and mustard. Spread 1 1/2 teaspoons mayonnaise mixture on bottom half of each bun; top each with 1 lettuce leaf, 1 tomato slice, and 1 patty. Divide onion mixture evenly among servings; top with top halves of buns.