How to Make It
Place allspice, peppercorns, and cardamom pods in a high-powered blender; process on high speed until spices are finely ground, about 30 seconds. Add scallions, ginger, chiles, garlic, 1/4 cup water, tamarind paste, thyme, soy sauce, and oil; process on high speed until mixture is smooth, about 1 minute. Reserve 3 tablespoons marinade for jerk brine; cover and chill remaining marinade until ready to use.
Stir together 3 tablespoons reserved jerk marinade, 2 cups water, and salt in a medium saucepan. Bring to a boil over high, and cook, stirring occasionally, until salt is dissolved. Remove from heat, and add ice. Let brine stand until ice is melted, about 20 minutes.
Place chicken quarters in a large plastic container. Pour vinegar over chicken; rub vinegar into chicken using lime halves. Discard limes and any leftover vinegar. Pour cooled jerk brine over chicken in container; press chicken down to submerge. Cover and refrigerate at least 4 hours or up to 6 hours. Remove chicken from brine; discard brine. Return chicken to container. Add remaining jerk marinade, and rub to coat chicken. Cover and chill at least 8 hours or up to 24 hours.
Rub excess marinade off chicken quarters; place chicken on a rimmed baking sheet. Sprinkle chicken with 1 tablespoon salt, and let stand at room temperature while smoker preheats.
Prepare smoker according to manufacturer’s instructions using 5 pounds pimento or pecan wood chunks. Bring internal temperature to 250°F to 275°F; maintain temperature for 15 to 20 minutes. Place chicken quarters on grate, skin side up, and smoke, covered, until a meat thermometer inserted in thickest portion registers 160°F, 1 hour and 30 minutes to 2 hours, adding wood as needed to maintain smoker temperature of 250°F to 275°F.
Remove chicken from smoker; place in a large roasting pan. Cover tightly with aluminum foil, and let chicken steam 30 minutes. Unwrap chicken, sprinkle with remaining 1/2 teaspoon salt, and serve.