Combine cider, salt and brown sugar in large pot. Bring to a boil, stirring, until salt and sugar dissolve. Remove from heat. Add garlic. Transfer to large bucket or non-reactive container large enough to hold turkey and brine. Stir in 16 cups (4 L) cold water and cool completely.
Remove giblets, gizzards and turkey neck and reserve in refrigerator for later use. Add turkey to brine, adding more water if needed to cover completely. Cover and refrigerate 12 to 14 hr. Remove turkey from brine and discard. Drain turkey on rack set on large baking sheet (about 15 min.), then dry inside and outside of turkey with paper towels.
Preheat barbecue to medium-high heat. Stir oil with thyme, rosemary and pepper. Rub mixture inside and outside of turkey. Place turkey on rack fitted in roasting pan, along with reserved neck, gizzard and giblets. Place roasting pan on grill. Close lid and cook, basting occasionally with pan juices and tenting with foil if browning too quickly. Maintain barbecue temperature at 425°F to 450°F. (220°C to 230°C) for 2 1/4 hr. to 2 1/2 hr., until meat thermometer inserted into thickest part of inner thigh registers 185°F (85°C).
Transfer turkey to cutting board and tent with foil. Let stand 20 to 30 min. before carving. Strain juices from roasting pan, skim off fat. Serve with sliced turkey.