1 Place the chicken onto a rotisserie and set the grill on high. 2  Cook for 10 minutes. 3  During that time, quickly mix together the butter, 1 tablespoon of salt, paprika and pepper. 4  Turn the grill down to medium and baste the chicken with the butter mixture. Turn the grill down to medium and baste the chicken with the butter mixture. 2  Close the lid and cook for 1 to 1 1/2 hours, basting occasionally, until the internal temperature reaches 180 degrees F (83 degrees C) when taken in the thigh with a meat thermometer.