1. Whisk dry rub ingredients in a medium-sized bowl until well combined. Set aside.
2. Remove gizzard and trim excess fat. Rinse chicken and pat dry with paper towels (including the cavity). Place chicken on a baking sheet and refrigerate uncovered for at least 20 minutes to dry while you prepare the smoker.
3. Set the grill to smoke and allow the fire to get rocking, 4 – 5 minutes. Turn the grill up to 225 degrees, and leave to preheat with the door closed for 15 minutes.
4. Remove the chicken from the refrigerator and rub it all over with 1/4 cup of olive oil. Next, season generously with dry rub, including seasoning the cavity, and then place the chicken in a grate, breast side down. Then put it into the smoker for 2-1/2 to 3 hours, or until the thermometer reads 160 degrees F when inserted into the thickest part of the breast.
5. Remove chicken from smoker, shred and set aside.