To marinate the chicken, place the chicken, carrots, celery, onion, garlic, and thyme in a large nonreactive bowl. Pour over the wine, cover, and marinate in the refrigerator overnight.

To get the garnishes ready, place the bacon into a cold skillet over low heat and cook, stirring occasionally, until lightly browned. With a slotted spoon, transfer the bacon to a plate.

Add the mushrooms to the fat in the pan and cook over medium-high heat, stirring occasionally, until lightly browned, about 5 minutes. If you need more fat, add 1 tablespoon of butter. Transfer the mushrooms to the plate with the bacon.

In a small saucepan, bring the stock to a boil. Add the pearl onions, reduce the heat to low, and simmer until the tip of a paring knife easily pierces the onions, about 20 minutes. With a slotted spoon, transfer the onions to the plate with the bacon and mushrooms. Reserve the stock.

To cook the chicken, remove the chicken from the marinade and pat dry with paper towels. Place a fine-mesh strainer over a large bowl and strain the marinade into the bowl; discard the vegetables.

Heat 2 tablespoons of the butter and the olive oil in a large Dutch oven over high heat until bubbly and very hot. Working in batches if necessary, add the chicken, skin side down, and cook until browned, 8 to 10 minutes. Turn and cook until lightly browned, about 5 minutes. Remove the chicken to a plate and sprinkle the flour into the pot.

Whisk in the reserved stock and marinade and bring to a rapid boil. Add the chicken legs, thighs, and wings to the pot, reduce the heat to low, cover, and cook, turning once, for 20 minutes.

Add the chicken breasts, bacon, mushrooms, and pearl onions and cook, uncovered, until the chicken is fully cooked, 25 to 30 minutes longer.