Position a rack in the center of the oven and heat the oven to 350ºF.
Bring a 2-quart saucepan of water to a boil. Drop in the onions and blanch them for 1 minute. Drain in a colander, then shower with cold water to stop the cooking. Peel the onions, leaving enough of the root end intact that they will remain whole while cooking.
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 over medium-high heat until very hot. Arrange the chicken pieces skin side down in the pan (it’ll be crowded), cover with a splatter screen, if you have one, 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 to a plate. Pour out and discard all the fat from the pan.
Put the pan over medium heat. Add the peeled onions, mushrooms, bacon, and carrot to the pan and cook until the bacon is crisp and the vegetables are browned, 8 to 10 minutes. 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 bring 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.
With a slotted spoon, transfer the chicken, onions, bacon, carrots, and mushrooms to a serving dish and keep warm by covering the dish loosely with foil. Discard the thyme sprigs. Tilt the sauté pan and skim off as much fat as possible from the sauce. Bring the sauce to a boil over mediumhigh heat, reduce the heat to medium low, and simmer until the sauce is slightly thickened, skimming off any skin that forms on top, about 5 minutes. Season to taste with pepper (the sauce should be nicely salted at this point). Pour the sauce over the chicken, sprinkle with the thyme leaves, and serve.
Serve with mashed potatoes.
Pair this braise with a crisp, minerally white from northern Italy, such as the Gini Soave Classico.