Use a large skillet with deep sides and a lid (you can also use a Dutch oven or a Braiser). Place skillet over medium-high heat and fry the diced bacon until crispy, about 15 minutes. Transfer the bacon to the paper towel-lined plate. Set aside.
Increase the heat under the skillet to high (careful for splatter!) and brown the chicken breasts, skin-side down, in the bacon grease until golden brown, 2-4 minutes. Flip chicken and brown the other side for another 2-4 minutes. Remove browned chicken to a plate and set aside.
Lower the heat under the skillet to medium-high. Add the carrots, onions, shallot, and mushrooms to the skillet and stir to coat with the bacon and chicken drippings. Sauté the veggies until golden and soft, about 15 minutes.
Carefully pour the red wine into the skillet, deglazing the bottom by scraping up any brown bits. Bring the wine to a boil and let simmer until it’s reduced by about 1/3, which should take about 5 minutes.
Pour the chicken broth into the skillet and add the thyme sprigs, then stir to combine. Nestle the chicken breasts back into the skillet among the veggies.
Cover the skillet and reduce heat to low. Let coq au vin simmer for 20-30 minutes, basting the chicken breasts every 10 minutes, until the chicken is cooked through.
Uncover the skillet, remove the thyme sprigs, and let the coq au vin continue to simmer for another 5 minutes. Stir the cornstarch mixture from earlier into the skillet, then let simmer another 2-3 minutes until the sauce has thickened.
Serve coq au vin immediately over noodles, mashed potatoes, polenta, etc. Garnish with cooked bacon pieces before serving.