Cut off the root end of each pearl onion and make an "x" with your knife in its place. Bring 2 to 3 cups of water to a boil and drop in the onions for 1 minute. Remove the onions from the pot, allow them to cool, and then peel. You should be able to slide the onions right out of their skin. Set aside.
Sprinkle the chicken on all sides with kosher salt and freshly ground black pepper. Place the chicken pieces, a few at a time, into a large (1- or 2-gallon) sealable plastic bag along with the flour. Shake to coat all of the pieces of the chicken. Remove the chicken from the bag to a metal rack set inside a sheet pan.
Add 2 tablespoons of the water and salt pork to a large, 12-inch saute pan set over medium heat. 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, along with 1/3 cup water and cook until the mushrooms collapse, 2 to 4 minutes. Add the butter, then cook, stirring occasionally, until the mushrooms are browned, 7 to 10 minutes.
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 tomato paste, onion, celery, carrots, garlic, thyme, bay leaf, and chicken stock. Add all of the remaining wine. Cover and refrigerate overnight.
The next day, heat the oven to 325ºF.
Place the chicken in the oven and cook until the chicken is tender, 2 to 2 1/2 hours. 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 set over a large bowl, removing the carrots, onion, celery, thyme, garlic, and bay leaf. Return the strained 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 30 to 45 minutes.
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. Serve over egg noodles, if desired.
Cook's 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.