Cut the skin into strips, and season the duck quarters lightly with salt, cayenne and white pepper.

Heat a large stockpot over medium high heat. Add the strips of duck skin and the duck quarters to the pot and cook until the fat is rendered (about 1/4 to 1/2 cup), and the duck is brown on all sides, about 5 to 7 minutes. Remove the skin and discard, and drain the duck pieces on paper towels.

Sprinkle the flour into the fat in the pan, stirring with a wooden spoon for approximately 15 minutes to make a brown roux. Add the onions, celery and bell pepper to the pot and cook until soft, about 3 minutes. Add the garlic to pan and cook another 2 minutes, then deglaze the pot with the beer, stirring well to remove any browned bits from the bottom of the pot. Add the stock, bay leaves, parsley and return the duck quarters to the pot, and simmer for 1 to 1 1/2 hours or until the duck meat is tender, skimming to remove any foam which forms on the surface.

In a frying pan, over medium high heat, cook the andouille until brown on all sides, stirring to turn, about 10 minutes. Set aside.

Preheat the oven to 400 degrees F.

Using tongs or a slotted spoon, carefully remove the duck pieces from the hot stock and cool before pulling the meat from the bones. Discard the bones and return the meat to the stockpot and add the cooked andouille. Continue cooking as needed until the etouffee is thickened and slightly reduced. Transfer the etouffee to a large casserole or skillet.

Place the puff pastry on a lightly floured board and roll out to fit the casserole dish. Carefully lay the puff pastry sheet on top of the dish, turning the edges under and crimping to seal. Brush the top of the sheet with the egg wash and bake in the oven until the top of the pastry is puffed and golden brown, about 10 to 15 minutes.