Place all marinade ingredients in a blender and process until smooth. Reserve about 1/2 cup for basting.
Place the chicken halves in a large zippered freezer bag and pour the marinade over them. Seal the bag and place in the refrigerator at least 4 hours, preferably overnight.
Preheat your gas grill (or covered charcoal grill) to medium-high heat. Initially, you will brown the chicken over direct heat and finish the cooking over indirect heat. On a gas grill, this means moving the chicken to the UNLIT side of the grill then putting the cover down. On a charcoal grill this means that you move the coals to the periphery of the grill and place the chicken in the center where there's no coals under it, then placing the cover on the grill.
Remove chicken from marinade and pat dry on paper towels. Discard used marinade.
Cook the chicken, skin side down, over direct heat for about 5 minutes or until browning starts. Turn chicken and move it to the unlit side of the grill. Cover and cook for about 45 minutes, turning every so often and basting with reserved marinade, until juices run clear or registers 165F/75C on an instant-read thermometer.
Tent with foil and let the chicken rest for 10 minutes.