Preheat the smoker to 250°.
Rub: Combine all of the ingredients and set aside.
Spritzer: Add the apple juice, cider, and water to a spray bottle and shake. Set aside.
Sauce: In a small saucepot add in the BBQ sauce, butter and 1 cup of the spitzer mix. Keep warm.
Rub the olive oil all over both sides of the chicken creating a thin layer.
Next, generously season the chicken on both sides with the rub. There will be 2-3 tablespoons leftover.
Add the chicken to the smoker and cook for 1 hour. Spray the chicken with 4-5 sprays each of the spritzer and do so after that every 20-25 minutes or until it reaches an internal leg temperature of 150° to 155°.
Sauce the chicken using a brush and smoke until it reaches an internal leg temperature of 165°.
Remove from the smoker and let rest for 20-30 minutes before serving.
Serve alongside BBQ sauce.