In a large bowl, season the chicken with salt and pepper. Toss until well coated.
Tie the thyme sprigs, bay leaves, and parsley stems together with kitchen twine to make a bouquet garni.
Add the bouquet garni to the bowl with the chicken. Pour the wine over the chicken. Cover with plastic wrap and refrigerate for at least 2 hours or, even better, overnight.
In a large Dutch oven, cook the bacon over medium-low heat until the fat has rendered and the bacon is crisp, 15 minutes. Using a slotted spoon, transfer the bacon to a paper towel-lined plate, leaving any excess fat in the pot.
Remove the chicken from the marinade, reserving the liquid, and place on a paper towel-lined sheet pan. Pat the chicken with more paper towels until completely dry.
Heat the bacon fat over medium heat until nearly smoking. Working in batches, add the chicken legs skin-side down in a single layer and cook until well browned, 5 minutes per side. Remove the chicken from the pot and set aside.
Add the carrots, onion, sliced mushrooms, and garlic to the pot. Cook until the vegetables are softened and lightly browned, about 10 minutes.
Stir in the tomato paste and cook for 3 minutes, until darkened and fragrant. Then, sprinkle in the flour and cook for another minute, until incorporated.
Pour the brandy into the pot and cook, scraping up any browned bits at the bottom of the pot, until the liquid has evaporated, 2 minutes.
Add the reserved marinade, including the bouquet garni. Stir to incorporate, then bring to a boil and cook until the liquid is reduced by half, about 15 minutes.
Return the chicken and half the cooked bacon to the pot. Cover, reduce the heat to low, and simmer for 1 hour, until the chicken is tender and the sauce has thickened.
Meanwhile, heat the olive oil and butter in a large, preferably nonstick skillet over medium-high heat. Add the pearl onions and a pinch of salt and sugar. Cover, reduce the heat to low, and cook for 15 minutes, shaking the skillet often to move the onions around.
Uncover, add the quartered mushrooms, and increase the heat to medium-high. Continue to cook until all the vegetables have softened, 5-8 minutes. Add the onions and mushrooms to the coq au vin.
Sprinkle with chopped parsley and the reserved bacon. Serve with crusty bread.
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