Spatchcock the turkey: Remove backbone and flatten the bird by pressing down on the breastbone. Pat completely dry with paper towels.
Mix the dry brine: Combine all dry ingredients in a small bowl and reserve 1 tbsp for later. Zest lemon and orange, stir in spice rub.
Season: Gently loosen the skin over the breast and thighs. Rub the mixture evenly over the entire bird, including under the skin and the back.
Dry brine: Place the turkey on a wire rack over a tray and refrigerate uncovered for 24–36 hours. This allows the salt and spices to draw out and reabsorb moisture, seasoning the meat deeply and helping the skin dry out for crispness.
Prepare your pit or smoker to cook at 300-325 degrees. Cook turkey to 150 degrees.
Melt butter, add Cajun seasoning, paprika, lemon & orange juice, and garlic. Brush on turkey when the breast reaches 150 degrees. Remove turkey when breast is at 160 degrees and thigh is at 175 degrees.
Rest for 30 minutes then apply another brush of glaze.