Bring 1 quart of water to a boil. Cut off the root end of each pearl onion, then use the very tip of your blade to inscribe a shallow "X" in its place. Blanch the onions in the boiling water for 1 minute. Remove, cool, and pop the skins right off the little buggers by applying pressure to the sides with your thumb and forefinger. Set aside.
Put the flour in a large (1- or 2-gallon) zip-top plastic bag. Season the chicken on all sides with salt and pepper. Dredge a few pieces at a time by putting them in the bag, sealing it, and shaking vigorously. Dancing helps. As the poultry is retrieved, place it on a wire rack set over a half sheet pan, and let it rest for 5 minutes.
Put the salt pork in a 12-inch saute pan along with the 2 tablespoons water. Cover and cook over medium heat until the water is cooked out the pork sticks are golden brown and crisp, 8 to 10 minutes. Remove the pork from the pan and set aside in a mixing bowl.
In the same pan, using the remaining fat, add the pearl onions, sprinkle with salt and pepper, and saute until lightly brown, 8 to 10 minutes. Transfer the onions to the mixing bowl with the pork. Next, brown the chicken pieces in the pan until golden brown on each side, working in batches if necessary to not avoid overcrowding. Transfer the chicken to a 7- to 8-quart enameled cast iron Dutch oven.
Cook the mushrooms to the same 12-inch saute pan, adding the butter only if needed. Cook until the mushrooms collapse, 2 to 4 minutes. Add the mushrooms to the bowl with the pork and pearl onions, cover with plastic wrap, and refrigerate until ready to use.
Pour off any remaining fat and deglaze the pan with about 1 cup of the wine. Pour this into the Dutch oven along with the stock, tomato paste, onion, celery, carrots, garlic, thyme, and bay leaf. Add all of the remaining wine. Cover and refrigerate overnight.
The next day, heat the oven to 325ºF.
Place the Dutch oven in the oven and cook, covered, until the chicken is tender, 2 to 2 1/2 hours, stirring every half hour. The liquid should barely simmer and, when done, the chicken should be very tender. Reduce the oven to 200°F when the chicken is done.
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 set over a large bowl, removing the carrots, onion, celery, thyme, garlic, and bay leafThe bay leaf can be discarded at this point.. Return the strained sauce to the pot, place over medium heat, and cook until it has reduced by one third. Depending on how much liquid you actually began with, this should take 30 to 45 minutes. 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.
Once the sauce has thickened, add the pearl onions, mushrooms, and pork, and cook until heated through, about another 15 minutes. Taste and adjust seasoning if necessary, remove from the heat, add the chicken and serve over egg noodles, if desired.