In a large saucepan over high heat, bring the 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 foam. Allow the mixture to cool to room temperature.
In an unused 5 gallon plastic bucket or a container deep enough to hold the turkey (we used a deep stockpot), combine 3 quarts (12 cups) of water, the oranges, ginger, cloves, bay leaves and garlic. Add the apple juice mixture and stir.
Remove and discard 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 make sure it is completely immersed. Refrigerate for 8 to 12 hours or up to 24 hours.
Follow your grill's instructions for using wood chips. Set up the grill for indirect grilling over medium heat (with charcoal, build fire on one side; for gas grill, turn off one side of the jets).
Remove the turkey from the brine and pat dry with paper towels. Tie the legs together with cotton string. Lightly brush the turkey with vegetable oil. Place on a roasting rack set inside a heavy-gauge foil pan. Place on grill over drip pan. Cover grill. When the wings are golden brown, after about 40 minutes, wrap them with aluminum foil to prevent them from burning. Brush the rest of the turkey with vegetable oil. When the turkey breast is golden brown, about 1 hour, cover the turkey with aluminum foil to prevent the skin from getting too brown. (Add more wood chips and hot coals as necessary, usually adding more coals after 1 hour of cooking time.)
Using an instant-read or meat thermometer cook until the internal temperature at the thickest part of the thigh is about 180 degrees F and the temperature of the breast is 165 degrees F. Figure about 12 to 14 minutes per pound, or about 2 1/2 hours for a 12-pound turkey.
Transfer turkey to a cutting board or platter, cover loosely with foil and let rest 20 minutes before carving. Pan drippings in foil pan may be used to make gravy.