1. The bacon: place the diced bacon in a cold, large Dutch oven or other heavy lidded braising pit (7-quart works well), set oven medium heat, and cook the bacon, sitting often with a slotted spoon, until well browned and crisp on the outside but with some softness remaining inside, 12 to 15 minutes. Transfer the bacon to a plate lined with paper towels. Set the pot with the rendered bacon fat aside of the heat.

2. Heat the oven to 325 degrees.

3. Dredging the chicken: rinse the chicken pieces with cool water and pat dry with paper towels. Season on all sides with salt and pepper. Spread the flour in a wide shallow dish ( a pie plate works well), and dredge half the chicken pieces on at a time, placing each one in the flour, turning to coat both sides, and then lifting and patting lightly to shake off any excess..

4. Browning the chicken: add 1 tablespoon of the butter to rendered bacon fat in the pot and place over medium-high heat. When the butter has melted, ease in the dredged pieces of chicken, skin side down, without crowding Sear on both side, turning once with tongs, until a deep golden brown crust forms, 7 to 10 minutes total. Transfer the chicken to a large platter to catch the juices. Dredge the remaining chicken pieces, and discard the flour. Add another 1 tablespoon butter to the pot. Sear the dredged chicken as you did the first batch, turning once with tongs, until golden, 6 to 9 minutes. The second batch of chicken pieces may brown faster; lower the heat as bit of the skin begins to burn at all. A thick ruddy rust will have formed on the bottom of the pot that will later contribute great depth of flavor to the sauce. Transfer the chicken to the platter and pour off the fat from the pot without discarding the tasty browned bits. Return the pot to medium heat.

5. The aromatics and braising liquid: add the remaining tablespoon of butter to the pot and melt it over medium heat. Add the onion and carrot and toss to coat the vegetables in the butter. Sauté, stirring once or twice with a wooden spoon, unit the vegetables are beginning to soften and are flecked with brown, about 5 minutes. The brown crust on the bottom of the pot will continue to cook and soften from the vegetable juices released into the pot. Add the tomato paste and stir to smear the paste through the vegetables. Add the cognac and bring to a boil to deglaze, scraping the pot with a wooden spoon to dislodge the precious crust. Simmer, stirring a few times, until the liquid is almost all gone. Raise the heat to high, add the red wine, garlic bay leaf, thyme and parsley, and bring to a boil. Lower the heat to medium-high and simmer rapidly until the wine reduces by about half, about 15 minutes. Stir in the reserved bacon and the stock and bring to a boil. Using a ladle, scoop out about ½ cup of braising liquid and set aside for later cooking the pearl onions.

6. The braise: add the chicken pieces to the pot in this order: place the legs, thighs, and wings, and the wing tips, back, neck, heart and gizzard (if using) in the pot first, then put the breast pieces on top of them, skin side down. (keeping the breast pieces on top protects them from overcooking and drying out.) pour in any juices that collected as the chicken sat, and bring to a simmer. Cover the chicken with parchment paper, pressing down so that the paper nearly touches the chicken and extends over the sides of the pot by about an inch. Cover with the lid, and place on a rack in the lower third of the oven to braise. After 15 minutes, turn the breast pieces over with tongs. At the same time, check that the liquid is simmering quietly. If not, lower the oven temperature by 10 or 15 degrees. Continue braising gently for another 45 to 60 minutes, or until the breasts and dark meat are fork-tender.

7. While the chicken braises, cook the garnish: if using fresh onions, bring a pot of water to a boil. Add the onions and bring the water back to a boil. Boil the onions for 2 minutes, drain, and rinse with cold water. Using a small paring knife, slip off the onion skins, and pat the onions dry. (If using frozen onions, skip this step: frozen onions have already been blanched and peeled.)

Heat 1 tablespoon of the butter in a large skillet (preferably a 12-inch nonstick) over medium heat. Add the blanched onions (or still-frozen onions) and sauté, stirring and shaking frequently, until tinged with brown, 3 to 4 minutes. Season with salt and pepper, add the reserved ½ cup of braising liquid, cover, and simmer, shaking the pan frequently, until the onions are tender when pierced with the tip of a knife, about 12 minutes (or 3 to 4 minutes if using frozen). Remove the lid, increase the heat to medium-high, and boil to reduce the liquid to a glaze. Transfer the onions and liquid to a small bowl, scraping the pan with a rubber spatula as best you can. Return the pan to medium-high heat and add the remaining 1 ½ tablespoons of butter. When the butter stops foaming, add the mushrooms, season with salt and pepper, and sauté briskly. The mushrooms may release a lot of liquid at first. Continue to sauté, stirring occasionally, until the liquid has evaporated and the mushrooms develop an attractive chestnut brown sear, about 10 minutes. Remove from the heat and return the onions and liquid to the skillet. Set aside.

8. The finish: When the chicken is fork-tender and pulling away from the bone, transfer the breast, thigh, leg and wing pieces of a deep platter or serving dish (discard the wing tips, back, neck, heart, and gizzard), and cover loosely with foil to keep warm. Let the braising liquid settle for a moment and then, with a wide spoon skim off as much surface fat as you can without being overly fastidious. Place the pot over high heat and bring the juices to a boil. Reduce the juices until thickened to the consistency of a vinaigrette, about 10 minutes. Remove and discard the bay leaf if you like.

Lower the heat, add the reserved onion-mushroom garnish, and heat through, about 4 minutes more. Spoon the sauce over the chicken pieces, sprinkle with the chopped parsley, and serve.