Combine the vinegar, rum, habaneros, onion, green onion tops, thyme, olive oil, salt, pepper, allspice, cinnamon, nutmeg, ginger, and molasses into a blender or food processor. Pulse until mostly smooth.
Place the chicken in a large freezer bag, or in a large non-reactive bowl or baking dish. Pour lime juice over the chicken and coat well.
Rub the jerk paste over the chicken pieces, coating evenly and thoroughly. Seal the bag or cover the chicken in the pan with plastic wrap.
Refrigerate for 3-4 hours.
Remove chicken from the marinade bag or pan. Put the remaining marinade into a small saucepan.
Bring to a boil, reduce heat and simmer for 10 minutes. Set aside to use as a basting sauce for the chicken.
While basting the chicken, reserve a little of the boiled marinade to serve with the chicken for those who like it extra saucey.
Preheat grill to 400 °F (204.44 °C). Add salt and pepper to chicken halves. Place chicken halves skin side down on the grill. Cover and cook for about 40-50 minute. Keep the grill temperature between 350 °F (176.67 °C) and 400 °F (204.44 °C), turning the chickens occasionally and basting with marinade, until the chicken is cooked through.
The chicken is done when the juices run clear, not pink, when a knife tip is inserted into both the chicken breast and thigh. If using a meat thermometer, the temperature should read about 165 °F (73.89 °C) for the breast and 170 °F (76.67 °C) for the thigh.
Transfer chicken to a large platter and tent loosely with foil. Let stand 15 minutes.
Cut chicken into pieces. Serve with black beans and rice, and your favorite Central American lightly fermented cabbage relish.
For even more flavor, marinate the chicken overnight or up to 36 hours before cooking.