Place the chicken pieces in a large bowl and rinse them with the white vinegar and lime juice, then rinse under cold water and pat dry with paper towels.
Add the sliced onion, half of the scallions, half of the bell peppers, garlic, ginger, ground allspice, smoked paprika, salt, and black pepper to the bowl with the chicken. Toss everything together with your hands so the chicken is well coated.
Cover the bowl and refrigerate for at least 1 hour, or up to overnight for a deeper flavor.
Remove the chicken from the marinade and shake off any excess vegetables, setting the marinated vegetables aside to use later in the gravy.
Heat the vegetable oil in your Dutch oven over medium high heat until it's shimmering. Working in batches so you don't crowd the pot, add the chicken pieces skin side down and sear for 4 to 5 minutes per side, until deeply golden brown.
Transfer the browned chicken to a plate and set aside. Do not clean the pot, since those browned bits add flavor to the gravy.
Reduce the heat to medium and add the reserved marinated vegetables to the same pot, along with the remaining scallions and bell peppers. Cook, stirring often, for 3 to 4 minutes until the vegetables begin to soften and turn fragrant.
Stir in the brown sugar and let it cook for about 1 minute until it turns slightly glossy and deepens in color. Add the browning sauce and stir to coat the vegetables evenly.
Return the seared chicken to the pot, nestling the pieces into the vegetables. Pour in the chicken broth and ketchup, then add the whole Scotch bonnet pepper, thyme sprigs, pimento berries, and sliced carrot.
Bring the liquid to a boil, then reduce the heat to low, cover the pot, and let the stew simmer gently for 35 to 40 minutes, stirring occasionally. Check the chicken with a meat thermometer to confirm it has reached an internal temperature of 165 degrees Fahrenheit.
Once the chicken is tender and fully cooked, remove the whole Scotch bonnet pepper, thyme stems, and pimento berries from the pot and discard them.
If you'd like a thicker gravy, uncover the pot and let it simmer for another 5 to 10 minutes to reduce. Taste the gravy and adjust the salt and pepper as needed.
Serve the chicken hot, spooned generously with gravy over rice.