Preheat smoker to 250 degrees F using mesquite, maple, or apple wood.
Combine the ingredients for the turkey dry rub in a small bowl.
Prep the turkey by removing from the brine and removing excess from patting dry with paper towels.
Add 2 teaspoons of the dry rub into the butter. Stir together to combine.
Using your hands gently separate the skin from the turkey breast.
Rub a portion of the herb butter under the skin on each breast.
Then rub the remaining butter on the top of the skin. If you find that the butter is sticking to your hands and not the turkey, use the back of a spoon to ensure a nice even layer of butter over the top of the turkey breast.
Evenly sprinkle more of the dry rub on top of the butter getting the top and sides of the breasts.
Place turkey breast, breast side up, directly on the grates of the smoker.
Cook for 3-4 hours until the turkey reaches an internal temperature of 160 degrees F.
Remove the turkey from the smoker, placing on a baking sheet or cutting board. Tent the turkey with foil and allow to rest for 15-20 minutes before slicing.
To slice, cut the breast down the center to separate the two sides, then using a sharp knife, cut following the bone to carve out the breast. Lay the breasts down on the cutting board and slice thinly to serve.