Preheat oven to 350°F and place a rack near the bottom.

In a large Dutch oven, add bacon and heat over medium heat. Cook bacon until crispy and fat has rendered, 8 to 10 minutes. Using a slotted spoon, transfer bacon to a paper towel lined plate, leaving the fat in the Dutch oven.

Increase the heat to medium-high and season the chicken pieces generously with salt and pepper. Working in batches, brown the chicken in the Dutch oven and sear to a deep brown on both sides, about 6 minutes, working in batches as needed. Transfer chicken to a baking sheet.

Add onion, carrot, celery, and leek to the Dutch oven and cook, stirring occasionally, until vegetables are browned, about 8 to 10 minutes. Add tomato paste and garlic and cook, stirring constantly, until tomato paste turns a deep brick red, about 1 to 2 minutes.

Add the wine to the Dutch oven and using a wooden spoon, scrape the bottom of the pan to release any browned bits. Cook until the mixture begins to thicken, 4 to 5 minutes. Return chicken to the Dutch oven along with any juices from the baking sheet and add enough stock to barely cover the chicken. Add thyme, bay leaves, and peppercorns, stirring to combine. Cover the Dutch oven and place in the preheated oven. Braise the chicken until fork tender, 45 to 55 minutes.

If using fresh pearl onions, blanch in rapidly boiling salted water until tender, 4 to 5 minutes. Drain in a colander and let cool. Trim ends and the peel should slide off easily. Set aside. If using frozen onions, thaw and drain of excess liquid.

To a large skillet set over medium-high heat, heat butter and oil. Add the onions and mushrooms and sauté until slightly browned. Drizzle with sherry vinegar and season liberally with salt and pepper. Place pan in oven with the Dutch oven roast, stirring occasionally until onions and mushrooms are golden brown, and most of the liquid has evaporated, 15 to 20 minutes. Stir in the reserved bacon bits and set aside.

Remove Dutch oven from oven and, using tongs, carefully transfer the chicken to a rimmed baking sheet. Cover loosely with foil to keep warm.

Strain the braising liquid through a fine mesh strainer into a large saucepan, pressing solids with a wooden spoon to release as much liquid as possible. Place saucepan over high heat and boil braising liquid until reduced and thickened, 10 to 12 minutes. Skim the surface with a ladle to remove any excess fat. Taste and adjust seasoning with salt and pepper.

To serve: Place one piece of chicken each into 8 individual shallow serving bowls and ladle sauce over each piece. Garnish with pearl onions and mushrooms and sprinkle with reserved bacon and thyme leaves. Serve immediately.