1. Bring apple juice to a boil in a heavy 3-qt. saucepan. Remove from heat, and stir in bourbon and next 3 ingredients, stirring until sugar and salt are dissolved. Cool completely (about 20 minutes); stir in ice.
2. Remove neck and giblets from chicken; reserve for another use. Place chicken and apple juice mixture in a 2-gal. zip-top plastic freezer bag; seal. Place bag in a shallow baking dish, and chill 4 hours, turning bag occasionally.
3. Remove chicken from brine, discarding brine; pat chicken dry with paper towels. Tie chicken legs together with kitchen string, and tuck chicken wingtips under.
4. Whisk together 2 Tbsp. brown sugar and next 2 ingredients in a large bowl. Add apples and next 2 ingredients, tossing to coat. Place mixture in a single layer in a lightly greased shallow roasting pan. Sprinkle with desired amount of kosher salt and freshly ground black pepper. Place chicken, breast side up, on top of apple mixture in pan.
5. Light one side of grill, heating to 350° to 400° (medium-high) heat; leave other side unlit. Place pan over unlit side, and grill, covered with grill lid, 45 minutes. Stir apple mixture. Grill, covered with grill lid, 1 hour and 15 minutes to 1 hour and 20 minutes or until a meat thermometer inserted into chicken thigh registers 165° and vegetables are tender. (Shield after 45 minutes to prevent excessive browning.) Remove from grill, cover chicken and vegetables with foil, and let stand 20 minutes. Transfer chicken to a serving platter. Toss apples and vegetables with pan juices to coat; sprinkle with parsley.