For the chicken and rub: In a smoker with hickory or other hardwood, bring the wood to smoke and maintain a temperature of 175 to 180 degrees F.
In a bowl, blend the paprika, salt, cayenne, cumin and garlic powder well. Next, season the chicken halves with the rub, both on the skins and in the interiors. Place in the smoker for 2 hours. The smoker temperature should be maintained between 165 and 170 degrees F for the cooking period. Remove the chicken from the racks and allow to rest until cooled, for a minimum of 2 hours.
For the sauce: In a bowl with a whisk, mix together the ketchup, vinegar, mustard and horseradish.
To reheat and glaze the chicken, preheat the grill. Evenly coat the skins of the chicken halves with the butter and salt. Grill until crisp, about 3 minutes. Flip and finish reheating for 4 minutes. During this period, glaze the skins with 2 tablespoons of the sauce.
If you desire to serve the chicken immediately from the smoker, brush the skins with the sauce during the final 30 minutes of the original 2-hour period.