Heat closed contact grill for 5 minutes.
When grill is heated, place bacon on bottom grill surface. (If necessary, cook 4 slices at a time.) Close grill; cook 3 to 4 minutes or until browned. Remove bacon from grill; drain on paper towels. Cut bacon slices in half crosswise.
With rubber scraper, carefully scrape most of drippings from grill into drip tray. Add onion to grill. Close grill; cook 4 to 6 minutes or until tender.
To make sandwiches, layer Cheddar cheese, cooked bacon, onion and mozzarella cheese between bread slices. Place 2 sandwiches on bottom grill surface. Close grill; cook 3 to 5 minutes or until bread is toasted and cheese is melted. Repeat with remaining sandwiches.