Roughly chop the first six ingredients for the jerk marinade, and toss them into a food processor. Add remaining ingredients and pulse until smooth, but chunky. Reserve 1/2 cup of the sauce in the refrigerator.
Make a series of 1/4-inch deep cuts into the chicken quarters to allow for the marinade to penetrate into the meat. Place them in a sealable plastic bag. Pour the marinade over the chicken and massage to ensure chicken is completely coated. Seal the bag and place the chicken in the refrigerator for at least 2 hours but preferably overnight.
Light your grill and set up for two-zone cooking. While the grill is preheating, remove chicken and remaining marinade from the refrigerator and allow them to sit at room temperature for no more than 20 minutes.
Place chicken over direct high heat and allow the skin to char but not burn. Flip and turn the chicken often until desired level of charring has happened, usually about 10 minutes.
Slide chicken over to indirect heat and using a long silicone basting brush, glaze the chicken with remaining marinade. Close the grill lid and allow chicken to cook until the internal temperature reaches 180°-185° F, usually about 20 minutes.
Remove chicken from the grill, garnish as desired, and serve hot. Authentic Caribbean jerk chicken is usually chopped into pieces before being served. Unless you have a heavy-duty meat cleaver this is challenging so I generally separate the leg and thigh and serve them like that, but you can serve them whole as well.
Notes
I often blog about recipes that I have adapted to the grill from my childhood, or about flavor profiles influenced by my travels during my time in the U.S. Marines. This recipe is neither. Many years back, I had the great fortune of spending a week in Jamaica on vacation. The place I stayed had an in-house chef who was amazing and one of the nicest ladies I've ever met.I asked her to teach me to make Jamaican jerk chicken, and she was kind enough to show me her way. What's even better, she taught me how to prepare and grill the chicken, so that was super cool. This was well before I started this site, but I've taken any chance I could to get behind the grill since I was a kid. Grilling chicken in Jamaica was a real treat.