Cut the duck skin into strips. Season the duck quarters lightly with salt, cayenne, and white pepper.
Heat a large heavy pot or Dutch oven over medium-high heat. Add the strips of duck skin and the duck quarters and cook, turning the duck once, until the fat is rendered and the duck is brown on both sides, 5 to 7 minutes for each side. Remove the skin with a slotted spoon and discard. Transfer the duck pieces to paper towels to drain.
Whisk the flour into the fat in the pot. Reduce the heat to medium and cook, stirring with a wooden spoon, to make a brown roux, for about 15 minutes. Add the onions, celery, and bell peppers and cook stirring often, until softened, about 4 minutes. Add the garlic and cook until fragrant, about 1 minute. Stir in the beer, scraping the pot to remove any browned bits from the bottom.
Add the stock, parsley, bay leaves, and duck quarters and bring to a boil. Reduce the heat to medium-low and simmer until the duck is tender, about 1 1/2 hours, skimming off any foam that forms on the surface.
Meanwhile, cook the andouille, stirring occasionally, in a skillet over medium-high heat until browned, about 10 minutes. Set aside.
Preheat the oven to 400?F.
Using tongs or a slotted spoon, carefully remove the duck pieces from the pot and let cool, then remove the meat from the bones, discarding the bones.
Return the meat to the pot, along with the andouille. Continue cooking until the ÈtouffÈe is slightly reduced and thickened. Transfer the ÈtouffÈe to a large casserole or ovenproof skillet. Cool for 10 minutes.
Roll out the puff pastry on a lightly floured work surface to a round 1/8-inch thickness, slightly larger than the casserole dish. Carefully lay the puff pastry sheet on top of the dish, turning the edges under and crimping them. Brush the top with the egg glaze.
Bake until the pastry is puffed and golden brown, about 15 minutes. Serve hot.
Yield : 6 servings