Smash the allspice as fine as possible using a mortar and pestle. Put in a food processor and add the seasoning salt, peppers, green onions, thyme and 1/2 cup water. Blend to the consistency of lumpy oatmeal and pour into a bowl.
Rub the chicken with the jerk seasoning and stuff the jerk seasoning between the skin and the meat. Place the chicken pieces into a storage pan and let marinade overnight (at least 12 hours).
Remove the excess seasoning from the chicken. Heat the grill to high heat (about 350 degrees F) and then place the chicken meat-side down first as to not let the chicken burn too quickly. Flip the chicken every 5 to 7 minutes until the skin has a nice brown color with grill marks, and then every 7 to 10 minutes on lower heat until cooked, around 40 minutes total or until the internal temperature of the chicken reaches 165 degrees F.
Chop each quarter into 4 to 5 pieces to get the real Jamaican jerk experience.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.