Prepare grill for indirect grilling at between 325 -400 degrees.
Combine paprika and next 5 ingredients in a small bowl for the rub. Rinse chicken and pat dry. Remove excess fat and skin from the chicken.
Sprinkle 2 tsp of the rub inside the chicken cavity. Brush outside of chicken with the olive oil, then sprinkle with the remaining rub, pressing to adhere.
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Pour 3/4 Dr Pepper into a measuring cup and set aside for the sauce. Poke 2-3 additional holes into the top of the can.
Arrange chicken cavity on cab. Pull chicken forward to form a tripod with the can serving as the rear leg.
Place the chicken over unlit portion of the grill; cover and cook until skin is golden and crisp, juices run clear, and an instant read thermometer inserted into the thickest part of the thigh registers 180, about 1 - 1 1/4 hours, rotating the chicken halfway through grilling.
Remove chicken from grill to a baking sheet; let rest for 15 minutes.
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