How to Make It
Combine 1/2 gallon water and 1 1/2 cups salt in a large saucepan. Cook over medium, whisking constantly, until salt dissolves, about 3 minutes. Pour mixture into a very large stockpot or other large container; add remaining 11/2 gallons water. Place turkey in brine. Place a plate upside down on top of turkey to submerge completely. Cover; refrigerate 12 hours.
Remove turkey from brine; discard brine. Pat turkey dry with paper towels, including cavity. Let stand 1 hour at room temperature. Stuff cavity with onion, garlic, lemons, and sage. Rub oil over turkey breasts and legs. Sprinkle with remaining 1 teaspoon salt. Tie legs together with kitchen twine; tuck wing tips under.
Open vent of smoker completely. Prepare a charcoal fire in smoker according to manufacturer’s instructions, and push coals to one side. Bring internal temperature to 300°F to 325°F; maintain temperature 15 to 20 minutes. Place 4 cups soaked applewood chips on coals. Position grate in smoker, and place turkey, breast side up, over side without coals. (Legs should be pointing toward coals.) Smoke turkey, covered with smoker lid, maintaining temperature inside smoker between 300°F and 325°F and brushing with vinegar every 45 minutes, until a thermometer inserted into thickest part of breast registers 155°F, 12 to 15 minutes per pound, approximately 2 hours and 30 minutes to 4 hours. Transfer turkey to a cutting board; let rest 30 minutes. Carve turkey, and serve.