Preheat the oven to 375 degrees F.
Sprinkle the chicken with salt and pepper. Allow to rest on the counter for up to 2 hours to come to room temperature.
Heat a large oven-safe pan (such as a cast-iron skillet) over medium-high heat and add the olive oil. Right as the oil begins to smoke, add the chicken, skin side down. Press gently while cooking and allow to sear until a peek beneath reveals golden brown skin, about 10 minutes. Using tongs, flip the bird over so that the skin is on top and place the pan in the oven. Be gentle with the tongs, they could rip the skin; try gripping the legs to flip. Place the pan in the oven and roast about 30 minutes.
Meanwhile, in a saucepot on medium heat, add the butter and melt. Add the onions, garlic and allspice, and sprinkle with a pinch of salt and a grind of pepper. Sweat the onions for a few minutes, but do not brown. Add the brown sugar, mustard, hot sauce and vinegar. Cook a bit more to bring the flavors together, and then remove from the heat.
When the chicken is around the 140 degree F mark, about 30 minutes in, remove from the oven and brush on the glaze. Return to the oven and continue to cook until an instant-read thermometer reads 165 degrees F, about 15 minutes more. Let the chicken rest 10 minutes before serving. Cut off the legs, and then slice the breast off and cut into portions.
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