Bring vegetable stock, salt, brown sugar, peppercorns, allspice berries, and candied ginger to a boil in large stockpot over medium-high heat, stirring occasionally to dissolve solids. Remove from heat, cool to room temperature, and refrigerate to 40°F.
In large non-reactive container, combine brine with cold water. Set turkey in brine, breast side down, placinga weight on top to keep turkey submerged if necessary. Place in refrigerator and let brine for 8 to 16 hours.
Remove turkey from brine and pat dry with paper towels. Place turkey on rack over rimmed baking sheet and allow to air-dry overnight in refrigerator. (This step is optional, but will result in a crisper skin.)
Combine apple, onion, cinnamon, and water in microwave-safe bowl. Microwave on high for 5 minutes. Add to turkey’s cavity, along with rosemary and sage. Close turkey cavity by threading wooden skewer through flaps of skin. Fold wings under body and tie legs together. Allow turkey to come to room temperature while you prepare grill.
Light one chimney full of charcoal. When all charcoal is lit and covered with gray ash, pour out and arrange coals on either side of charcoal grate. Place foil pan between two piles of coals to capture drippings to serve with turkey or to make gravy. Place turkey on rotisserie and cook over medium heat until instant read thermometer registers 155°F in thickest part of the breast, 2 to 3 hours. Remove turkey from rotisserie and let rest for 20 minutes, carve and serve.