In a large stockpot, large enough to hold all the pigeon pieces in 1 layer, heat 2 tablespoons butter over medium-high heat until it foams and subsides and add the onion, celery and carrot. Cook over medium-high heat until the vegetables are soft and browned, about 5 minutes.
Add the pigeon pieces and cook over high heat until the meat is nicely browned, about 10 minutes. Season with salt and pepper and add the wine. Cook over high heat until the wine evaporates completely, then add a little of the chicken stock and cook over medium-high heat. Meanwhile, spread the slices of bread with the remaining 2 tablespoons butter.
Preheat the oven to 450 degrees F.
Remove the pigeon pieces from the pot and remove all the bones. Shred the meat and add the reserved livers and 1 liter of broth. Soak 2 of the slices of bread in the broth and lay them along the bottom of a casserole. Sprinkle with some of the cheese and add some of the meat, liver, and stock mixture. Repeat with the remaining bread, 2 slices at a time, cheese, and meat mixture. The top layer should be cheese.
Place the casserole in the oven and cook for about 1 hour, adding 1/4 cup of the remaining chicken stock every 15 to 20 minutes. Serve in bowls with additional cheese grated over, if desired.
In a large, heavy-bottomed saucepan, heat the oil over high heat until smoking. Add all the chicken parts and brown all over, stirring to avoid burning. Remove the chicken and reserve. Add the carrots, onions, and celery to the pot and cook until soft and browned. Return the chicken to the pot and add 6 quarts of water, the tomato paste, peppercorns, and parsley. Stir with a wooden spoon to dislodge the browned chicken and vegetables bits from the bottom of the pan. Bring almost to a boil, then reduce heat and cook at a low simmer until reduced by half, about 2 hours, occasionally skimming excess fat. Remove from heat, strain, and press on the solids with the bottom of a ladle to extract out all liquids. Stir the stock to facilitate cooling and set aside. Refrigerate stock in small containers for up to a week or freeze for up to a month.
Recipe copyright 2001, Mario Batali. All Rights Reserved.