In a food processor, puree the scallions, garlic, allspice, thyme, salt, brown sugar, jalapeño, oil, and lime juice until smooth, scraping down the sides halfway. Place the chicken in a gallon-sized plastic bag.
Pour the mixture over the chicken and seal the bag, pressing as much air out as you can. Turn the bag a few times to move the marinade around, coating the chicken.
Refrigerate for at least 1 hour or up to 24 hours, flipping the bag halfway through to make sure the chicken marinates evenly.
Preheat your gas grill with all burners on high and close the lid. When the grill is hot (after about 15 minutes), turn the 2 burners closest to the front to low, keeping the back burner on high.
Remove the chicken from the bag and arrange on the grates over the low heat burners. Cover and cook until the underside of the chicken is just beginning to develop light grill marks, 6 to 9 minutes. Flip the chicken, cover, and continue to cook until the chicken is firm to touch, 6 to 9 minutes longer.
To get a nice char on the chicken, now move the chicken to the part of the grill over high flames and cook uncovered until each side has solid grill marks, 5 to 6 minutes, flipping halfway.
The chicken should be completely firm to the touch and register 160 degrees F on an instant-read thermometer. Transfer the chicken to a plate, tent with foil, and let it rest for 5 minutes before serving.