Combine 2 quarts water, 1/2 cup bourbon, sugar, and kosher salt in a large Dutch oven, and bring to a boil, stirring until salt and sugar dissolve.
Add ice water and next 7 ingredients (through lemon), and cool to room temperature.
Add chicken to brine; cover and refrigerate 18 hours, turning chicken occasionally.
Soak wood chips in water for 1 hour; drain.
Remove the chicken from brine; pat chicken dry with paper towels. Strain brine through a sieve; discard brine and reserve 2 apple quarters, 2 lemon quarters, 2 onion quarters, and garlic. Discard remaining solids.
Sprinkle chicken cavity with pepper; add reserved solids to chicken cavity. Lift wing tips up and over back; tuck under chicken. Tie legs.
Remove the grill rack, and set aside. Prepare the grill for indirect grilling, heating one side to high and leaving one side with no heat. Pierce the bottom of a disposable aluminum foil pan several times with the tip of a knife.
Place pan on heat element on heated side of grill; add 1 cup 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.
Let chips stand for 15 minutes or until smoking; reduce heat to medium-low. Maintain temperature at 27
Coat the grill rack with cooking spray; place on grill.
Place chicken, breast side up, on grill rack over foil pan on unheated side.
Combine the remaining 1 tablespoon bourbon and butter; baste chicken with the bourbon mixture. Close lid, and cook for 2 hours at 275 or until thermometer inserted into meaty part of thigh registers 16
Add remaining 1 cup wood chips halfway through cooking time.
Place chicken on a platter; cover with foil.
Let stand for 15 minutes. Discard the skin before serving.
