1. Rinse chicken and pat dry. Place chicken, breast side down, on a cutting board. Using poultry shears, cut along both sides of backbone and open chicken like a book. Reserve backbone for stock or discard. Turn chicken breast side up and, using the heel of your hand, press firmly against breastbone until it cracks. Tuck wing tips under so they don't burn on grill. Place chicken in a large rimmed baking pan.
2. In a food processor, combine remaining ingredients and pulse to make a thick paste. Rub 1 cup paste on both sides of chicken, then cover pan with plastic wrap and chill 1 hour. Set aside remaining paste.
3. Prepare grill for indirect medium heat (350° to 450°; you can hold your hand 5 in. above cooking grate only 5 to 7 seconds). If using charcoal, light 60 briquets on firegrate. When coals are covered with ash, about 20 minutes, divide into banks on opposite sides of firegrate and let burn to medium. The area over the section cleared of coals is the indirect-heat area. If using gas, turn all burners to high, close lid, and heat 10 minutes. Then turn off one of the burners and reduce heat for other burner(s) to medium. The area over the turned-off burner is the indirect-heat area.
4. Place chicken, skin side down, over direct heat; cover grill and cook until chicken is browned, about 8 minutes. Turn chicken over and move to indirect heat. If using charcoal, add 12 more briquets to fire to maintain medium heat. Continue cooking chicken, covered, turning as needed for even browning, until a thermometer inserted in thickest part of breast to the bone registers 160°, 25 to 35 minutes more. Remove chicken from grill and let rest 5 minutes, then cut into quarters and serve with reserved paste.
Note: Nutritional analysis is per serving.