For the chicken: Season the chicken with salt and pepper. Rest on the counter for 2 hours to come to room temperature.
Preheat a grill to 400 degrees F. Lightly oil the grill grates.
Grill the chicken over indirect heat, rotating occasionally, until the internal temperature reaches 165 degrees F, about 50 minutes.
For the glaze: While the chicken is grilling, melt the butter in a small pot. Add the onion, allspice, a pinch of salt and black pepper. Cook until the onion is tender, then add the garlic and cook until tender but not browned. Add the brown sugar, mustard, vinegar and hot sauce. Reduce about 5 minutes, then remove from the heat.
When the chicken is nearly cooked through, brush on the glaze. Cover the grill and continue to cook until the glaze begins to color and the chicken is done. Cover the chicken loosely with foil and rest for 15 minutes to allow the juices to redistribute. Then cut into sections and serve.
Adapted from "Sunny's Kitchen" by Sunny Anderson. Copyright Clarkson Potter 2013. Provided courtesy of Sunny Anderson. All rights reserved.