Preheat the oven to 450°F.

Remove the neck and any innards from the cavity of the duck and rinse innards and duck well under cool running water. Pat dry.

Using a sharp boning knife, cut the legs away from the duck and then cut each leg quarter in half at the joint. Cut the wings off of the duck, then carefully cut the breast meat away from the bones. Cut each breast half into two pieces. Place the 2 legs, 2 thighs and 4 portions of breast meat onto a platter and season with the Rustic Rub. Cover with plastic wrap and refrigerate while you make the duck stock.

Cover a large baking sheet with aluminum foil and drizzle the 2 teaspoons of vegetable oil over the foil. Cut the duck wings apart at the joints and place them on the baking sheet. Cut the duck carcass into several pieces using poultry scissors and place on the baking sheet. Add the reserved neck, gizzards, and heart to the baking sheet and season everything lightly with salt and pepper. Transfer to the oven and roast until everything is golden brown, 40 minutes to an hour, turning as necessary to promote even browning.

Place the roasted duck pieces into a large stockpot and add the water, quartered onions, celery ribs, smashed garlic, 2 of the bay leaves, thyme sprigs and peppercorns. Bring to a boil over high heat, reduce the heat to a simmer, and cook, skimming any scum from the surface, until you have a rich stock, 2 to 3 hours.

Remove from the heat and strain the stock through a fine-mesh sieve. Discard all solids. Set stock aside while you make the roux.

Heat the remaining cup of oil over medium high heat in a large, heavy Dutch oven until hot. Whisk in the flour and cook, stirring constantly, until the roux turns the color of a dark copper penny. Should the roux begin browning too quickly, reduce the heat so that it doesn't burn. As soon as the roux is the correct color, immediately add the chopped onions, bell peppers, celery, and garlic and cook, stirring, until the vegetables are tender, about 6 minutes. Slowly whisk in the stock and add the remaining 2 bay leaves, the dried thyme, and season lightly with salt and cayenne. Bring the liquid to a boil and then reduce the heat to a simmer.

While the liquid is coming to a simmer, heat a large nonstick skillet over high heat. When hot, add the seasoned duck pieces, skin-side down, and cook until golden brown and crispy. Brown on the second side and then add the duck pieces to the gumbo.

To the drippings remaining in the skillet, add the mushrooms, in batches if necessary, and season lightly with salt and pepper. Saute until the mushrooms are golden brown and fragrant. Add the mushrooms to the gumbo and cook the gumbo until any floury taste is gone and the duck is tender and falling from the bone. Serve the gumbo with each guest getting a piece of duck or, if preferred, remove the pieces of duck from the gumbo and take the meat from the bones. Stir the duck back into the gumbo and serve over hot white rice, garnished with green onions.