Soak wood chips in water 1 hour; drain well.
Loosen skin from thighs and drumsticks by inserting fingers, gently pushing between skin and meat. Rub Kansas City Dry Rub evenly under loosened skin; let stand at room temperature for 30 minutes.
Remove grill rack; set aside. Prepare grill for indirect grilling, heating one side to medium-high and leaving one side with no heat.
Pierce bottom of a disposable aluminum foil pan several times with the tip of a knife. Place pan on heated side of grill; add half of wood chips to pan. Place another disposable aluminum foil pan (do not pierce pan) on unheated side of grill. Pour 2 cups water in pan. Coat grill rack with cooking spray; place grill rack on grill.
Place chicken on grill rack over foil pan on unheated side. Close lid; cook 1 1/2 hours. Add additional wood chips halfway during cooking time. Turn chicken over; cover and cook 30 minutes or until a thermometer registers 165°. Remove chicken from grill; let stand 10 minutes. Remove and discard skin.
Bring sauce to a simmer in a small saucepan. Brush chicken with 1/2 cup sauce. Serve chicken with remaining sauce.