In a large saucepan over high heat, bring apple juice, brown sugar, and salt to a boil, stirring to dissolve the sugar and salt. Cook for 1 minute, remove from heat, and skim off the foam. Allow the mixture to cool to room temperature.
In a 5 gallon plastic bucket or other container large enough to easily hold the turkey, combine 3 quarts of water, the oranges, ginger, cloves, bay leaves, and garlic. Add the apple juice mixture and stir.
Remove and discard the fat from the turkey cavity. Reserve the neck and giblets for another use. Rinse the turkey inside and out, drain, and submerge the turkey in the brine. If necessary, top with a heavy weight to be sure it is completely immersed. Refrigerate for 24 hours.
Follow the grill's instructions for using wood chips (soak chips in water for at least 30 minutes prior). Set up the grill for indirect cooking over medium heat.
Remove the turkey from the brine and pat with paper towels until very dry. Tie the legs together with cotton string. Lightly brush the turkey with vegetable oil, and place on a roasting rack set inside a heavy-gauge foil pan. Cook indirectly over medium heat. When the wings are golden brown, about 40 minutes, wrap them in aluminum foil to prevent them from burning. Brush the rest of the turkey with vegetable oil. When the turkey breasts are golden brown, about one hour, cover the turkey with aluminum foil to prevent the skin from getting too brown. The turkey is done when its juices run clear, the internal temperature is bout 180 degrees F, and the internal temperature of the breast is about 165 degrees F. Figure 12 to 14 minutes per pound.
Transfer the turkey to a cutting board or platter, cover loosely with aluminum foil and let rest 20 minutes before carving. The pan drippings may be used to make gravy.