For the brine: Add the salt, sugar, allspice, chile, garlic, ginger and 1 quart water to a large pot and stir to combine. Place over medium-high heat, bring to a boil and cook until the salt and sugar have dissolved, about 5 minutes.
Pour the brine over the ice in a large bowl and let cool to room temperature. Add the chicken to the brine, cover and refrigerate for 24 hours.
For the jerk paste marinade: Stir together the soy sauce, Worcestershire sauce, tamarind paste, scallions, thyme, minced garlic, minced ginger, allspice, brown sugar, salt, cinnamon, cloves, bay leaves and chiles in a medium bowl. Reserve 5 tablespoons for the Jerk BBQ Sauce and set aside.
Remove the chicken from the brine; discard the brine. Pat the chicken dry, then toss to coat with the jerk paste marinade, slightly massaging it into the chicken. Cover and refrigerate for 24 hours.
For the jerk BBQ sauce: Heat the oil in a medium saucepan over medium-low heat until shimmering. Add the onions, garlic, ginger and the reserved 5 tablespoons of Jerk Paste Marinade and simmer until the onions are translucent, 5 to 8 minutes. Add the ketchup and brown sugar and summer over low heat until the sauce is deep red and caramelized, about 30 minutes. Remove the sauce from the heat and transfer to a blender. Blend until smooth and set aside.
To cook the chicken: Preheat the oven to 325 degrees F. Heat a well-seasoned cast-iron grill pan over medium heat until very hot, about 10 minutes, then add some oil to the pan. Working in batches, remove the chicken from marinade (reserve the marinade), place the chicken on the grill pan skin-side down and cook until well charred, flipping them halfway through, about 10 minutes. Transfer the chicken to a rimmed baking sheet and drizzle with the reserved marinade. Continue with the remaining chicken. Transfer the baking sheet the oven and cook until the chicken is tender but not falling off the bone, about 45 minutes. Turn the broiler on for the last minutes and broil until the skin crisps.
Remove the chicken from the oven, brush it with some of the Jerk BBQ Sauce and let it rest 10 to 15 minutes. Serve with the remaining Jerk BBQ Sauce on the side.
Wear gloves while handling the Scotch bonnet chile and avoid touching your eyes.