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Add the 2 tablespoons of water to a large, 12-inch saute pan over medium heat along with the salt pork. Cover and cook until the water is gone, and then continue to cook until the salt pork cubes are golden brown and crispy, approximately 8 to 10 minutes. Remove the salt pork from the pan and set aside.
In the same pan, using the remaining fat, add the pearl onions, sprinkle with salt and pepper, and saute until lightly brown, approximately 8 to 10 minutes. Remove the onions from the pan and set aside. Next, brown the chicken pieces on each side until golden brown, working in batches if necessary to not overcrowd the pan. Transfer the chicken into a 7 to 8-quart enameled cast iron Dutch oven.
Add the mushrooms to the same 12-inch saute pan, adding the 1 tablespoon of butter if needed, and saute until they give up their liquid, approximately 5 minutes. Store the onions, mushrooms and pork in an airtight container in the refrigerator until ready to use.
Pour off any remaining fat and deglaze the pan with approximately 1 cup of the wine. Pour this into the Dutch oven along with the chicken stock, tomato paste, quartered onion, carrots, celery, garlic, thyme, and bay leaf. Add all of the remaining wine. Cover and refrigerate overnight.
The next day, preheat the oven to 325 degrees F.
Place the chicken in the oven and cook for 2 to 2 1/2 hours, or until the chicken is tender. Maintain a very gentle simmer and stir occasionally.
Once the chicken is done, remove it to a heatproof container, cover, and place it in the oven to keep warm. Strain the sauce in a colander and remove the carrots, onion, celery, thyme, garlic, and bay leaf. Return the sauce to the pot, place over medium heat, and reduce by 1/3. Depending on how much liquid you actually began with, this should take 20 to 45 minutes.
Once the sauce has thickened, add the pearl onions, mushrooms, and pork and cook for another 15 minutes or until the heated through. Taste and adjust seasoning if necessary, remove from the heat, add the chicken and serve. Serve over egg noodles, if desired.
Cooks Note: If the sauce is not thick enough at the end of reducing, you may add a mixture of equal parts butter and flour kneaded together. Start with 1 tablespoon of each. Whisk this into the sauce for 4 to 5 minutes and repeat, if necessary.