In a large bowl, combine the chicken pieces, vegetable oil, onion, garlic, ginger, thyme, allspice, cinnamon, nutmeg, and Scotch bonnet peppers. Mix well to coat the chicken evenly with the spices. Cover the bowl with plastic wrap and marinate in the refrigerator for at least 30 minutes or up to 24 hours.
Preheat a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the chicken pieces and cook until browned on all sides, about 5 minutes per side.
Add the chicken broth, diced tomatoes, brown sugar, salt, and pepper to the pot. Stir to combine.
Bring the mixture to a boil, then reduce the heat to low and cover the pot. Let the chicken simmer for 30 minutes.
Add the carrots and bell peppers to the pot and stir to combine. Cover the pot and let the chicken simmer for an additional 15 minutes, or until the vegetables are tender.
In a small bowl, mix together the cornstarch and water to create a slurry. Add the slurry to the pot and stir to thicken the sauce.
Serve the Jamaican stew chicken hot, garnished with fresh cilantro.