Pat the duck dry with paper towels, and season on all sides with the cayenne and salt as needed. (NOTE: If using wild duck, see the discussion on dry-aging on pages 228–229.)
Put the flour in a shallow dish. Dust the duck pieces with the flour and shake off excess. In a heavy Dutch oven or stockpot, heat the oil over medium heat, and, working in batches, if necessary, sear the duck pieces on all sides for several minutes, or until a rich brown. Transfer the duck to a large plate and set aside.
Pour off all but 3 tablespoons of fat from the pot, add the andouille, and brown lightly. Transfer to the plate with the duck. Stir in the bell peppers, onion, celery, and garlic, and cook until the onions are translucent and softened but not brown. Stir in the tomatoes, bay leaves, and 2 cups of the stock, and bring to a boil over high heat, scraping bottom of the pot to incorporate any browned bits.
Return the duck and andouille to the pot, along with any accumulated juices on the plate. Bring to a boil over high heat, then reduce the heat to a simmer. Partially cover, and cook until the duck is just tender, 1 hour and 15 minutes. Stir in the rice and the remaining 2 cups stock, and bring to a boil. Reduce the heat to low, cover, and simmer until the rice is tender and the grains have absorbed most of the liquid, about 20 minutes. Discard the bay leaves. Add the Worcestershire and season with salt and hot sauce to taste. Top with the green onions, and serve immediately.