Position a rack in the bottom third of the oven and heat the oven to 400°F.
Bring 1 quart of water to a boil in a 2-quart saucepan. Add the onions and cook for 1 minute. Drain in a colander, then rinse with cold water to stop the cooking. Peel the onions, leaving enough of the root end intact that they will remain whole during cooking.
Remove the giblets from the chicken and discard or save for another use. Pat the chicken dry and trim away any excess fat. Cut off the first two joints of the wings and reserve them. Season the inside of the chicken with salt and pepper. Truss the chicken. Season the outside of the chicken with salt and pepper.
Heat a 5- to 6-quart Dutch oven or other heavy-duty pot over high heat. Add the olive oil and heat until shimmering hot. Add the chicken, breast side down. Lower the heat to medium high and cook, undisturbed, until nicely browned. Turn the chicken with tongs, being careful not to tear the skin, and brown on the remaining sides, about 10 minutes total. Transfer to a rimmed baking sheet.
Return the pot to medium-high heat. Add the carrots, mushrooms, cipollini, and reserved wings. Cook, stirring occasionally, until lightly browned, about 3 minutes. With a slotted spoon, transfer the vegetables and wings to the rimmed baking sheet with the chicken. Discard the oil from the pot, add the wine, raise the heat to high, and cook until reduced by half, about 10 minutes. Add the chicken broth and thyme and return the chicken to the pot. Bring the liquid to a simmer, cover, and braise in the oven for 20 minutes. Take the pot out of the oven and increase the oven temperature to 500°F.
Add the vegetables and wing tips to the pot, return it to the oven (do not cover), and cook until an instant-read thermometer inserted in the breast registers 165°F, about 15 minutes. Transfer the chicken to a cutting board and tent loosely with foil. Transfer the vegetables to a baking dish, cover, and keep warm. Discard the wing tips and thyme.
Pour the braising liquid into a glass measuring cup or fat separator and let sit for 5 to 10 minutes. Discard the fat and pour the liquid into a 2-quart saucepan. Boil over medium-high heat until reduced to about 1-1/2 cups, about 15 minutes. Season to taste with salt and pepper.
Carve the chicken into breast and leg pieces. Divide the chicken and vegetables among four shallow bowls and pour some of the sauce over all. Sprinkle with the parsley or chervil and serve.