Heat chicken stock in a small pot to a low simmer, then set aside over low heat.
Add pancetta to a large low-sided pot or rondeau over medium heat and cook, stirring occasionally, until pancetta is crisp, about 5 minutes. Remove pancetta from the pot and drain the fat, reserving 1 tablespoon of the fat. Add olive oil and heat to shimmering. Add the mushrooms and sear on all sides until golden brown and crisp, 8 to 10 minutes. Season with salt. Add 1 tablespoon red wine vinegar to deglaze the pot. Remove mushrooms from pot and set aside. Add a drizzle of olive oil to the same pot on high heat, then the celery, onions, carrot and chestnuts. Cook until they begin to brown, about 10 minutes. Add the garlic and the tomato paste and cook until richly caramelized but not burnt, another 5 to 7 minutes. Add the red pepper flakes and cook for 1 minute, then deglaze with 2 cups of the chicken stock and add the reserved mushrooms. Cook until reduced by three-quarters, about 10 minutes. Season with thyme, salt, pepper and the remaining 1 1/2 tablespoons red wine vinegar and turn off the heat. Add 2 more cups of the hot chicken stock to the pot along with the bread, pancetta and the reserved pancetta fat.
Let stand for 45 minutes, allowing the bread to absorb all of the flavor.
Preheat the oven to 350 degrees F.
If the stuffing seems dry, add the remaining cup chicken stock. Add the egg and fold to combine. Transfer to a 9-by-13-inch baking dish. Top with Parmesan and bake until a nice crust forms on top, about 25 minutes. Garnish with chives.