Optional: Pat the chicken thighs dry and place on a sheet pan, uncovered. Refrigerate to allow the skin to dry out for 1 to 4 hours. This isn’t necessary, but it does lend to a crispier skin.
Position the rack to the top and preheat the oven to 375℉. Pat the chicken dry (if you didn't do this already) and season all over with salt and pepper.
Heat a large, oven-safe skillet (preferably a 3.5 quart-braiser) over medium heat. Add the pancetta and cook, stirring, until it's cooked through and crisp, 5 to 6 minutes. Using a slotted spoon, transfer to a paper towel-lined plate and set aside. Discard all but 2 tablespoons of the fat in the pan.
Increase the heat to medium-high and, working in batches, arrange the chicken thighs, skin side down, in a single layer and cook until a golden-brown crust forms, 6 to 8 minutes. Transfer the browned chicken to a plate and set aside. Repeat with the remaining chicken.
Reduce the heat to medium, add the carrots and shallots, and continue to cook, stirring, until they begin to soften, about 3 minutes. Add the garlic and mushrooms and continue to cook, stirring, being careful not to burn the garlic, until the mushrooms begin to soften, another 3 minutes.
Sprinkle in the flour and cook, stirring, until it is well combined and begins to toast, 1 to 2 minutes.
While stirring, slowly pour in 1 cup of Champagne until it is well incorporated, scraping up any browned bits on the bottom of the pot. Slowly stir in the remaining 1/2 cup of Champagne. Add a small pinch of salt and a few cracks of black pepper. Bring to a gentle simmer, 3 to 4 minutes.
Add the cooked pancetta back to the sauce. Using kitchen twine, tie the thyme and rosemary sprigs into an herb bouquet and toss into the sauce along with the bay leaf. Nestle the chicken, skin side up, into the sauce.
Transfer to the oven on the top rack and bake, uncovered, until the chicken is cooked through and very tender, and the skin is golden brown, about 45 minutes.
Remove from the oven and let rest for 5 minutes to allow the juices to settle. Transfer the cooked chicken to a clean plate and stir in the heavy cream and mustard until well combined. Return the chicken to the pot and nestle into the sauce.
Garnish with fresh parsley. Serve over mashed potatoes and/or with crusty bread to soak up the sauce. Enjoy!