Heat grill. Brush onion slices and steaks with oil. When ready to grill, place onion on gas grill over medium-high heat or on charcoal grill 4 to 6 inches from medium-high coals. Cook 5 minutes.
Turn onion; place steaks on grill. Cook onion and steaks 6 to 9 minutes or until onion is tender and steaks are of desired doneness, turning steaks once and topping with cheese during last minute of cooking time.
In small bowl, combine mayonnaise and horseradish; mix well. Spread bread slices with mayonnaise mixture. Top 4 slices with steaks and onion slices. (If desired, separate onion into rings.) Place remaining bread slices, mayonnaise side down, over onion.