Rinse chicken pieces and pat dry. Season all over with salt and pepper.
In a large oven-proof skillet (I use a French Oven), cook diced bacon over medium heat until most of the fat is rendered and pieces are browned.
Remove bacon with a slotted spoon, leaving the fat in the pot, and drain on a paper towel.
Add the chicken pieces to the pot and brown the skin in the bacon fat over medium-high heat. Meanwhile, roughly chop carrots, onions and celery into 1/2 inch pieces.
Remove chicken from pot and set aside. Add vegetables to the pot, using a little olive oil as needed to cook the vegetables. Saute vegetables for about 5 minutes until they begin to soften.
Add a cup of wine to deglaze the pan, that is, to loosen all the brown bits on the bottom. Scrape the bottom of the pan to get all those flavorful bits up.
Add the chicken pieces, the remaining wine, the stock, bay leaf, and the thyme leaves to the vegetables. Stir to combine, then cover with a lid.
Simmer coq au vin gently, over low heat, for about 40 minutes or until the chicken is tender and falling off the bone. (cooking time will vary with size and quality of chicken.)
Remove the pot from the heat and, using a sturdy pair of tongs, take out chicken pieces and set aside once again. Strain the sauce through a fine mesh strainer and discard the vegetables.
Return the sauce to the pot and bring to a simmer. Cook for 10 minutes at a low boil until it is reduced by half.
In a small bowl, mix together just 2 Tablespoons soft butter and flour. Using a whisk, incorporate the paste, called a 'Beurre manié', into the sauce. Mix well until all the lumps are gone. Cook the sauce another five minutes, whisking occasionally, until it is glossy and coats the back of a spoon. Taste for seasoning and add salt and pepper as needed.
In a separate saute pan, heat the remaining tablespoon of butter, and add mushrooms. Stir for a minute, then add a few tablespoons of water. This helps the mushrooms to release their juices. Cook over medium-high heat for another 5 minutes until the liquid is evaporated and mushrooms begin to brown.
Return the chicken to the sauce in the pot, and add the bacon and sauteed mushrooms. Stir together and heat gently. Serve over hot buttered egg noodles or atop boiled, smashed potatoes. Garnish with chopped fresh parsley.