a rotisserie grill attachment, kitchen twine and a disposable drip pan
Make the brine: Heat 1 cup water, 1/4 cup salt, the brown sugar, garlic, coriander seeds, 1 1/2 teaspoons paprika and 1/2 teaspoon cayenne in a small saucepan over low heat, stirring, until the salt dissolves. Let cool completely.
Put the chicken in a large resealable plastic bag. Pour in the brine and buttermilk; seal the bag. Refrigerate at least 4 hours or overnight, turning once or twice.
About 30 minutes before grilling, remove the chicken from the brine and pat dry. Let sit at room temperature.
Prepare a grill with a rotisserie attachment according to the manufacturer’s instructions. Preheat the grill to medium and prepare for indirect cooking: On a gas grill, turn off the center burner(s); on a charcoal grill, bank the coals to the sides.
Combine the butter, vinegar, ground coriander, garlic powder and the remaining 1/2 teaspoon paprika and 1/8 teaspoon cayenne in a saucepan. Cook over low heat, stirring, until the butter is melted. Set aside.
Truss the chicken with kitchen twine: Tie the legs together and tie the wings close to the body so the chicken is a round shape. Once the grill registers 325˚ F to 350˚ F, slide the chicken onto the rotisserie spit. Insert the prongs on the rod into the chicken so it’s snug; secure with the thumbscrews. (If the twine loosens, tie again.) Place the rod onto the rotisserie with a drip pan underneath; turn the rotisserie on.
Cover the grill and cook, basting the chicken with the butter mixture every 30 minutes, until a thermometer inserted into the thigh registers 165˚ and the skin is browned and crisp, 2 to 2 1/2 hours. (If using a charcoal grill, add coals as needed to maintain a temperature of 325˚ F to 350˚ F.)
Turn off the rotisserie. For a gas grill, turn off the grill and let the chicken rest on the rod for 10 minutes before carving. For charcoal, lay the chicken, on the rod, on a foil-lined baking sheet and let rest 10 minutes before carving.
Bank the coals to the sides and put a drip pan in the middle so it will catch any juices from the chicken and prevent flare-ups.