Remove the turkey from the refrigerator 45 to 60 minutes before cooking. Pat every surface thoroughly dry with paper towels — especially the skin folds, the inner cavity, and beneath the breast skin flap. Any pooled surface moisture will steam in the smoker and prevent crisping.
Prepare the compound butter: in a small bowl, combine the softened butter, thyme, rosemary, lemon zest, and a pinch of kosher salt. Mix until uniform. Work the compound butter under the loosened breast skin and spread it evenly over the breast meat in a thin layer. Rub any remaining butter over the exterior skin of the thighs.
4 tablespoons unsalted butter, softened to room temperature, 1 teaspoon fresh thyme leaves, roughly chopped, 1 teaspoon fresh rosemary, minced, 1 teaspoon lemon zest, 1 pinch kosher salt
Prepare your smoker or grill for indirect cooking at 325°F. For a charcoal kettle, arrange coals to one side or use the snake method, placing a drip pan filled with 1 cup of water directly beneath where the bird will sit. Add 3 to 4 apple wood chunks and 2 to 3 cherry wood chunks directly on or beside the hot coals. Allow the smoker to reach and stabilize at 325°F before adding the turkey.
Place the spatchcocked turkey skin-side up on the smoker grate over the drip pan, away from direct heat. Insert a leave-in probe thermometer horizontally into the thickest part of the breast, entering from the neck end and avoiding bone contact. Close the lid.
Smoke at 325°F, maintaining thin blue smoke. Do not open the lid for the first 60 minutes. Check wood chunk combustion if the smoke goes white or heavy — adjust intake vents to increase airflow and thin out the smoke to a clean, pale blue.
Beginning at 90 minutes, use a spot-check instant-read thermometer to probe the thigh at the thickest point, inserting from the inner thigh side toward the joint. Check both the breast and thigh temperature every 20 to 25 minutes from this point onward. The breast and thigh will not finish simultaneously — monitor both independently.
Remove the turkey from the smoker when the breast registers 157°F and the thighs register 175°F. A 12 to 14 pound spatchcocked turkey at 325°F typically reaches these temperatures in 2 hours to 2 hours 30 minutes, but use temperature, not time, as your pull trigger. If the breast hits 157°F before the thighs reach 175°F, tent the breast loosely with foil and continue cooking until the thighs catch up.
Transfer the turkey to a large cutting board. Tent loosely with foil — do not wrap tightly, as steam will soften the skin. Rest for 20 to 30 minutes. Carryover cooking will bring the breast to approximately 160 to 162°F and the thigh to approximately 178 to 180°F during this window.
Carve and serve. Separate the legs and thighs first by cutting through the hip joint, then remove the drumstick from the thigh at the knee joint. Slice the breast by running the knife along the breastbone and slicing crosswise against the grain at a slight angle.