Place patties on grill rack over medium heat. Cover grill; cook 12 to 15 minutes, turning once, until meat thermometer inserted in center of patties reads 160°F. Add bread slices to side of grill for last 5 minutes of grilling, turning once, until lightly toasted. Top patties with cheese. Cover grill; cook about 1 minute longer or until cheese is melted. Place patties on bread; top with onions.