Step 1In a medium saucepan, bring the onion, carrot, celery, 1 clove garlic (peeled), peppercorns and wine to a boil and simmer 5 minutes, and then cool the marinade completely.
Step 2Rub each piece of chicken with a pinch each of salt and pepper. Pack the pieces in a deep, non-metallic bowl and pour the cooled marinade and flavorings over them. Spoon the olive oil over to keep the chicken moist. Cover with plastic wrap and let the chicken marinate in the refrigerator for at least a day, turning the pieces from time to time, and up to 3 days if you like a full-bodied flavor of wine.
Step 3Remove the chicken pieces from the marinade and pat them dry with paper towels. Strain and reserve the marinade liquid, keeping the vegetables separate. Heat the oven to 325 degrees. Cut the bacon into lardons -- small cubes or sticks. If you're using thick-sliced bacon, cut the bacon crosswise into thin sticks.
Step 4Heat the oil in a braising pan and fry the lardons until browned and the fat runs. Transfer the lardons to a bowl using a slotted spoon and set aside for the garnish.
Step 5Add the chicken pieces to the pan, skin-side down, and saute over medium heat until well browned, at least 10 minutes. Turn, cook until the other side browns, 3 to 5 minutes, and remove them. Do not overcrowd the pan; if necessary, fry the chicken in two batches.
Step 6Add the reserved vegetables from the marinade to the pan and fry until they start to brown, 5 to 7 minutes. Stir in the flour and cook over high heat, stirring until it browns, 2 to 3 minutes. Pour in the marinade liquid and bring to a boil, stirring constantly until the sauce thickens. Simmer 2 minutes, then stir in the broth with the shallots, the remaining two garlic cloves (chopped) and bouquet garni.
Step 7Return the chicken to the pan, pushing the pieces down under the sauce. Cover the pan and cook in the oven, turning the chicken occasionally, until the pieces are tender and fall easily from a two-pronged fork, about 40 minutes to 1 hour. Some pieces may be done before others -- if so, remove them so they do not dry out from additional cooking, and continue cooking the rest.
Step 8Meanwhile cook the garnish. Melt the butter in a frying pan, add the onions, sprinkling one-eighth teaspoon salt and a pinch of pepper, and brown them over medium heat, about 5 to 7 minutes. Shake the pan from time to time so they color evenly. Lower the heat, cover and cook the onions, shaking the pan occasionally, until just tender, 8 to 10 minutes more. Remove them with a slotted spoon and add to the reserved lardons. Add the mushrooms to the pan, sprinkling a pinch each of salt and pepper and add a little more butter if needed. Saute until tender, 3 to 5 minutes. Add them to the lardons and onions.
Step 9When the chicken is cooked, remove the pieces from the sauce and set them aside. Strain the sauce into a bowl, discarding the vegetables and seasonings. Use a ladle to skim any fat on the surface. Wipe out the pan and add the garnish. Stir in the sauce; if it seems thick, add a little more broth. If it's too thin, reduce over high heat. Taste and adjust the seasoning. Add the chicken pieces, pushing them well down into the sauce, and heat gently 3 to 5 minutes so the flavors blend. Coq au vin improves if you keep it, covered, in the refrigerator at least a day and up to 3 days so the flavor mellows before serving.
Step 10To serve, reheat the chicken with the garnish and sauce on top of the stove. Transfer the chicken pieces to a serving dish or individual plates and spoon over the garnish with a little sauce. Sprinkle the chicken with chopped parsley and serve any remaining sauce separately.