Place the chicken in a large bowl and pour the wine overtop. Let marinate at room temperature while you cook the bacon. Preheat the oven to 350°F.
In a large Dutch oven or heavy-bottomed pot or high-sided skillet with a tight-fitting lid, add the bacon and cook over medium heat until golden and slightly crisp, about 8-10 minutes. Use a slotted spoon to remove the bacon to a paper towel and leave the fat in the pot. Increase the heat to medium-high. Remove the chicken from the wine marinade (reserving the wine for later use). Pat the chicken dry with a paper towel and season all over with 1 teaspoon of salt and black pepper. Add the chicken skin-side down to the pot, and brown for about 5 minutes. Turn the chicken and brown on the other side for about 5 minutes. Transfer chicken to a large plate.
Add the onion to the pot and cook over medium-high heat for about 5 minutes, breaking it up a bit and stirring occasionally. Then add the carrots, season with remaining ½ teaspoon of salt, and sauté for about 5 minutes. Add the garlic and tomato paste and stir while cooking until it becomes fragrant, for about 1 minute. Add the bacon back to the pot with the sprigs of thyme, bay leaf, and reserved wine, and bring to a simmer. Stir and scrape the bottom of the pan to help release the fond (aka all the yummy bits). Stir in the chicken stock and add the nestle the chicken back into the sauce. Cover with the lid and place in the oven to cook for 40 minutes.
While the chicken is cooking, sauté the mushrooms. Add 2 tablespoons of the butter to a large skillet and heat over medium. Once the butter has melted, add the mushrooms, season with ⅛ teaspoon of salt, and sauté, stirring occasionally until golden brown, about 7-8 minutes.
Remove the chicken from the oven. In a small bowl, mash the remaining 2 tablespoons of the softened butter with the flour, then stir in 2 tablespoons of the coq au vin sauce from the pot to make a paste. Remove the chicken from the pot and add the butter and flour mixture into the sauce, stirring while the butter melts and incorporates into the sacue. Add the chicken back into the pot along with the mushrooms, and simmer over medium heat for about 10 minutes to thicken the stew. Serve the coq au vin over buttery parsley noodles, mashed potatoes, or creamy polenta, and top with parsley.
Use a whole chicken that's been cut into pieces, or any combination of skin-on, bone-in chicken breasts, thighs, or legs.