Divide chicken into parts. Pat dry with paper towels, then generously season with salt and pepper, set aside.
Heat a Dutch oven over medium-high heat. Add bacon and saute until browned and crisp. Use slotted spoon to remove bacon. Place on a paper towel-lined plate to drain.
Add 2 tbsp of oil to bacon drippings and heat through. Add chicken in a single layer. Cook until well browned on all sides, about 8 minutes. Transfer chicken to a plate and set aside. Add carrot, shallots, celery and garlic to pan and saute until soft, about 3-5 minutes. Return chicken to pan and heat through. Add cognac to pan and heat through. Carefully ignite cognac, shake pan back and forth until flames subside.
Add wine and enough stock to cover chicken. Add tomato paste and herbs. Bring to a simmer, then cover and slowly simmer until chicken is fully tender when pierced with a fork, about 30 minutes. Remove chicken and set on a warmed platter. Simmer the sauce, uncovered,until has reduced slightly, about 15 minutes.
While the sauce is simmering, heat a medium saute pan over medium-high heat; add the remaining 1 tbsp of oil and heat through. Add pearl onions and cook, stirring, until onions are brown on all sides. Remove from pan and add to simmering sauce.
Return saute pan to heat; add butter and let it melt. Add mushrooms and a sprinkle of salt. Saute until mushrooms have released their fluid, it evaporates and the mushrooms have browned. Add mushrooms to the sauce.
When sauce has sufficiently thickened, add bacon, then season to taste with salt and pepper. Return chicken to pot and submerge the pieces in the sauce.
Simmer for 4 to 5 minutes to heat chicken through. Remove and discard herbs. Serve directly from pot.