Start the brining process tonight and thank us tomorrow!
For the chicken, in a large pot, bring 1 qt. water, the salt, sugar, garlic, allspice, bay leaves, and peppercorns to a simmer, stirring until the salt and sugar dissolve. Remove the pan from heat. Stir in 3 qt. cold water. Let the brine cool completely. Add the chicken. Cover and refrigerate overnight. Drain and pat dry. Transfer to a container and cover. Pack with ice in a cooler.
For the sauce, in a large wide saucepan, cook the onions, garlic, chili powder, oregano, smoked paprika, and sweet paprika over medium heat, stirring often, until the onions are soft and the mixture is aromatic, about 5 minutes. Stir in the vinegar, tomato puree, oyster sauce, mustard, brown sugar, and molasses. Simmer over medium-low heat, stirring occasionally, until the sauce thickens and reduces by about a third, about 30 minutes. Transfer 2 cups of the sauce to a jar and cover. Reserve for serving at the table. Place the remaining sauce in a small jar and cover. Reserve for brushing on the chicken while grilling.
Prepare a grill for medium heat. For a charcoal grill: When the charcoal is covered with ash, mound the charcoal on one side of the grill. For a gas grill: Leave one section unlit. Place the top rack on the grill and scrub it clean with a wire brush or a wad of foil.
Place the chicken on the hot side of the grill. Cook, brushing often with the sauce and flipping occasionally to coat both sides. Move to the cooler side of the grill as needed if the skin begins to burn. Cook until the chicken is cooked through or an instant-read thermometer registers 165° when inserted into the thickest parts of the chicken, 25 to 30 minutes. Serve with the reserved sauce.