Heat a large nonstick skillet over medium-high heat. Add oil; swirl to coat. Add onion; cook 5 minutes or until tender, stirring frequently. Remove onion from pan; do not wipe pan.
While onion cooks, combine beef, salt, and pepper; shape into 8 (1/2-inch-thick) patties. Add patties to drippings in pan. Cook 4 minutes on each side or until done, topping each patty with 1 cheese quarter during last 2 minutes of cooking time. Remove from pan. Add 4 rolls, cut sides down, to drippings in pan. Cook 1 minute or until toasted. Repeat procedure with remaining 4 rolls.
Place patties on bottom halves of rolls; top patties evenly with onion. Cover with top halves of buns. Serve with desired toppings.
Tip: You can season the beef and shape the patties in the morning; then when you get home they're ready to go.