Soak the wood chips or hickory wood blocks for at least a half hour. Place the drained wood chips in a disposable foil container. Wrap tightly with foil and poke a few holes on the top. Place directly on the heat of the grill.
Preheat the grill to 500 degrees Fahrenheit.
While grill preheats, rinse and pat dry the brined turkey. Spray a half-size aluminum foil steam pan with non-stick cooking spray. Rinse and pat dry the turkey breast and place in the pan.
In a bowl mix together the softened butter, brown sugar, chili powder, salt, cumin, oregano, mustard powder, and smoked paprika. Mix until well blended.
Then place some of the seasoned butter underneath the skin of the turkey breast. Then slather the rest of the butter mixture all over the outside of the turkey breast. Place aluminum foil over top and transfer to the grill.
Before placing the turkey breast on the grill, turn off one side of the burners. You will cook the turkey not directly over the heat source--in indirect heat. Open up a little piece of the aluminum foil and pour 2 cups of water into the pan. Close the foil back and close the lid on the grill.
Cook for 1 hour, remove the foil and baste the turkey with the juices in the foil pan. Close the lid and cook for an additional hour. Repeat again, and baste. Take the temperature of the turkey. It should read around 120 degrees Fahrenheit. Close the lid and cook for another 30-60 minutes or until the internal temperature reaches 165 degrees Fahrenheit.
Place the aluminum foil pan with the turkey in it on a baking tray and transfer into your home. Remove it out of the liquid and let rest for 10-15 minutes before slicing and enjoying.