Combine ground beef, garlic, salt and pepper in medium bowl, mixing lightly but thoroughly. Lightly shape into 12 half-inch-thick mini patties.
Place patties on grill over medium, ash-covered coals. Grill covered for 4 minutes. Turn and baste patties with 1/4 cup barbecue sauce. Continue grilling 4 to 5 minutes until internal temperature reaches 160 degrees. Sliders should not be pink in the center and juices should show no pink color. Place a cheese slice on each burger during last minute of cooking time.
Place burgers in cocktail buns and serve with additional barbecue sauce, as desired. Combine coleslaw mix with dressing and serve alongside the sliders and apple slices