In a large bowl or gallon-sized ziplock bag, combine the onion, bell pepper, scallions, garlic, brown sugar, paprika, allspice, ginger, and salt and pepper. Mix well with your hands or a utensil. Pat the chicken pieces completely dry with paper towels, then add them to the bowl along with the browning sauce. Massage the marinade thoroughly into each piece, ensuring every surface is well coated.

Cover the bowl or seal the bag and refrigerate for at least 2 hours, or preferably overnight (8+ hours). If marinating overnight, give the chicken a good massage at least once to redistribute the marinade evenly.

Remove chicken from marinade, shaking off large vegetable pieces but it's fine if some marinade clings to the chicken. Reserve all the marinade mixture. Heat oil in a large heavy-bottomed pot over medium-high heat until shimmering. Working in batches to avoid crowding, sear the chicken pieces on all sides until a deep brown crust forms, about 2-3 minutes per side. Transfer browned chicken to a clean plate and repeat with remaining pieces, adding 1-2 tablespoons more oil between batches only if needed.

Reduce heat to medium and add all the reserved marinade mixture to the pot. Sauté for 2-3 minutes, scraping up any caramelized bits from the bottom.

Return all browned chicken to the pot, nestling pieces into the marinade along with any collected juices. Add carrots, tomato sauce, fresh thyme, bay leaves, whole scotch bonnet pepper, and chicken stock. Stir gently to combine everything well.

Reduce heat to medium-low, cover with a lid, and let the chicken stew for 1½ to 2 hours until the meat is fall-off-the-bone tender. The chicken should be so tender it nearly shreds when touched with a fork.

Remove and discard the thyme stems, bay leaves, and scotch bonnet pepper. If you prefer a thicker gravy, increase heat to medium-high and cook uncovered for 5-10 minutes, stirring occasionally, until the liquid reduces by about one-third. Taste and adjust seasoning with additional salt and pepper as needed.