Trim excess fat and pat the chicken dry with paper towels, then season all over with kosher salt and black pepper and dust evenly with 1 tablespoon flour; let rest 15 minutes at room temperature.

Heat a 4–6 quart Dutch oven over medium-high and add 2 tablespoons vegetable oil; when shimmering, sear chicken skin-side down in batches until deep golden-brown, 4–6 minutes, then flip and brown 2–3 minutes; transfer to a tray.

Reduce heat to medium, add chopped onion and sliced carrots to the pot and cook until softened, about 4 minutes; add garlic, tomato paste, Scotch bonnet, thyme, allspice, and brown sugar and cook 1 minute; pour in 1 cup chicken broth and scrape the pot bottom to deglaze.

Return chicken to the pot skin-side up, add crushed tomatoes and soy sauce, bring to a gentle simmer, then reduce heat to low and cover partially; simmer gently for 30–35 minutes until the thickest part reads 165°F and meat is tender.

Remove the Scotch bonnet and thyme stems, then off the heat whisk in 2 tablespoons cold butter until the sauce becomes glossy; alternatively for a dairy-free finish, whisk 1 tablespoon oil with a cornstarch slurry (1 teaspoon cornstarch dissolved in 2 tablespoons cold water), stir into the hot sauce, and simmer 1–2 minutes to activate the starch and thicken.

Let the chicken rest 5 minutes in the sauce before serving and garnish with chopped cilantro or parsley if desired.