Remove the skin of the thawed turkey. Should be able to just peel off. May need to cut some portions off.
Place the turkey in a large resealable plastic bag. Place this in a large pot. This will help keep the turkey and brine from spilling.
In a small saucepan add one cup of water and the salt. Stir while heating until the salt has completely dissolved. You do not need to heat to boiling, just until salt is dissolved.
Pour the warm salt water over the turkey, into the bag. Add in the rest of the water, apple juice, sliced apples, lemons, rosemary sprigs, and whole black peppercorns.
Seal the bag and place the turkey, in the pot, into refrigerator and let sit overnight.
Next day, prepare grill/smoker according to directions. Preheat to 275.
Remove the turkey from the brine. Pat dry.
Cut the turkey breasts from the bones. Cut as close to the bones as possible to get the most meat possible.
Brush the turkey breasts with the melted butter. Sprinkle with the dry rub. Press into the butter to have as much rub adhere to the turkey as possible.
Place the turkey breasts, skin side (where the skin was) up.
Cook at 275℉ until turkey reaches 160℉ degrees.
Remove from the smoker. Allow the turkey to rest for about 15 minutes before cutting.
Slice and serve warm.