Season the chicken with salt and pepper. Let rest on the counter for 2 hours to come to room temperature. Preheat the oven to 375°F.
Pour the olive oil into a large, ovenproof pan over high heat. As the oil begins to smoke, add chicken, skin-side down. Press gently, especially on the legs, and sear until a peek beneath reveals golden brown skin, about 10 minutes. Using tongs, flip over so the skin is on top and place the pan in the oven. Roast until the internal temperature is around 140°F, 40 to 45 minutes.
For the glaze, in a small pot over medium heat, melt the butter. Add the onion and allspice and a pinch of salt and a few grinds of pepper. Cook until the onion is tender, then add the garlic and cook until tender but not browned. Add the brown sugar, mustard, hot sauce and vinegar. Reduce for about 5 minutes, then remove from the heat.
Remove the chicken from the oven and brush on the glaze. Return the chicken to the oven and continue to cook until the internal temperature reaches 165°F, about 20 minutes more.
Remove chicken and cover loosely with foil. Let rest for 15 minutes to allow the juices to redistribute. Cut into sections and serve.