In a large mixing bowl combine all the marinade ingredients. Remove 1/4 cup marinade, plastic wrap and place in fridge. Submerge 1 piece of chicken in bowl at a time and gently smear the marinade into both the skin and the flesh. Plastic wrap bowl and let marinade set for at least 1 hour (or overnight).
Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn off other burner(s). Adjust primary burner (or, if using 3-burner grill, primary burner and second burner) as needed to maintain grill temperature around 350 degrees. This is called indirect cooking.
Clean and oil cooking grate with swath of paper towel dipped in oil. Place chicken, skin side down, on cooler side of grill (not over flame). Place grill lid down and cook for 20 minutes. Alternate chicken pieces keeping skin side down, so that pieces that are closest to edge are now closer to heat source and vice versa. Cover and continue to cook until chicken registers 185 degrees F to 190 degrees F (85 degrees c), approximately 15 to 20 minutes.
Move all chicken, skin side down, to hotter side of grill and cook until skin is nicely charred (about 5 minutes). Flip chicken and cook until flesh side is lightly browned (1 to 2 minutes). Transfer to a clean platter, tent with aluminum foil, and let the chicken rest for 10 minutes.