Make a brine for the turkey: In a large stockpot combine the chicken stock, onions, garlic, and brown sugar. Bring to a boil over high heat. Remove the pot from the heat and let the brine cool completely. Place the turkey in a large roasting bag or a clean cooler or other large container, and carefully pour the brine into the container or bag. Seal the container or the tie the bag. Refrigerate, allowing the turkey to sit in the brine at least 8 hours, or preferably overnight.
When you are ready to cook the turkey, preheat a smoker to 250ºF.
Remove the turkey from the brine. Discard the brine. Pat the bird dry thoroughly with paper towels or a clean kitchen towel. Apply the rub all over the bird, inside and out.
Set a roasting rack inside a deep aluminum pan with the handles facing down, so the bird will be raised. Pour the apple juice into the bottom of the pan, underneath the meat rack. Place the turkey on the rack. Place the pan in the smoker and cook for 5 hours, or until the breast meat of the turkey reaches an internal temperature of 165ºF.
Remove the pan from the smoker. Allow the turkey to rest, loosely covered with foil, for 30 minutes. Carve the turkey and serve immediately.