Position a rack in the center of the oven and heat the oven to 350.F (175.C)
Season the chicken pieces generously with salt and pepper. In a straight-sided 10 or 11-inch ovenproof sauté pan with a lid, heat the oil and butter over medium-high heat. Arrange the chicken in the pan (you may have to do this in batches for even browning), and cook until deeply browned, about 5 minutes. Turn the pieces over and cook until the other sides are deeply browned, 3 to 5 minutes more.
Transfer the chicken to a plate. Next add the sliced mushrooms to the pan and cook until browned, season with a little salt and pepper. Remove from the pan and discard any remaining oil. Add the bacon to the pan and cook until crispy, then add in the shallot and carrot and cook for about 5 minutes more.
Add the wine and thyme sprigs and bring to a boil, scraping the browned bits from the bottom of the pan with a wooden spoon. Boil until the wine has reduced to 1/2 cup, about 5 minutes. Add the chicken broth and return to a boil.
Return the chicken pieces to the pan, along with any accumulated juices, and cover. Transfer to the oven and braise until the chicken is fork-tender and the drumstick meat starts to come away from the bone, 45 to 50 minutes.
Transfer the chicken to a serving dish and keep warm by covering the dish loosely with foil. Strain the onions, bacon, carrots, and mushrooms into a bowl using a fine mesh strainer and discard the thyme sprigs. Return the liquid to the sauté pan and skim off as much fat as possible from the sauce. Bring the sauce to a boil over medium high heat, reduce the heat to medium low, and simmer until the sauce is slightly thickened, skimming off any skin that forms on top for about 5 minutes.
Add in the heavy cream, Dijon mustard and fresh thyme leaves and bring back to a simmer. Once thickened, season to taste with pepper (the sauce should be nicely salted at this point). Add the veggies and bacon back into the sauce. Pour the cream sauce over the chicken, sprinkle with parsley, and serve.
Serve over mashed potatoes, or any side of your choice.