Stir together 1/2 cup apple butter, next 3 ingredients, and, if desired, liquid smoke in a medium bowl. Crumble ground chuck into bowl; gently combine ingredients until blended. Shape mixture into 4 (5-inch) patties.
Core apple, and slice into 1/2-inch-thick rings. Cut onion into 1/4-inch-thick slices, keeping rings intact. Brush apple and onion slices evenly with remaining 1 cup apple butter.
Coat cold cooking grate with cooking spray; place on grill over medium-high heat (350° to 400°). Arrange patties, apples, and onions on cooking grate, and grill, covered with grill lid, 4 to 6 minutes on each side or until patties are no longer pink in center and apples and onions are tender. Top each patty with 1 cheese slice during last 2 minutes of cooking.
Meanwhile, coat cut side of each roll with cooking spray, and place on grill, cut side down, around outer edge of cooking grate; grill 1 to 2 minutes or until golden brown.
Place patties on bottom halves of rolls; top with grilled apples, onions, and roll tops.
James's tip: Use your hands to combine meat mixture and to shape into patties. This keeps the burgers tender and moist.