Mix salt, allspice, brown sugar, onion powder, garlic, nutmeg, black pepper, ginger, cayenne pepper, cinnamon, and thyme in a large bowl; add chicken pieces and toss to coat with seasoning mix. Cover and marinate in the refrigerator, 2 to 4 hours.

Preheat the oven to 350 degrees F (175 degrees C). Pour vegetable oil into a 9x13-inch baking dish.

Place chicken pieces skin-side up into the baking dish. Bake in the preheated oven until juices run clear and chicken is no longer pink near the bone, 1 hour 20 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).

Turn the oven on to broil, and cook until the skin crisps, 2 to 5 minutes.