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1. 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.
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2. Soak wood chips in water for 1 hour; drain.
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3. 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.
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4. 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 275u00b0.
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5. 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 275u00b0 or until thermometer inserted into meaty part of thigh registers 165u00b0. 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.