Saute the pancetta/bacon. In a 10¼-inch cast iron skillet or other large oven-proof skillet, add the chopped bacon or pancetta and cook until crisp over medium heat, about 5 mins. Once crispy, remove the bits set aside. If more than 2 tablespoon of fat remains, remove the fat and leave 2 tablespoon in the pan. Cook chicken. (Skip to the next step if using cooked chicken) If using raw chicken, dice into cubes, add to the pan, and saute until cooked through about 5-7 mins. Remove from the pan and set aside.
Saute the veggies. Reduce heat to medium-low, and add 1 tablespoon of butter (skip if using cooked chicken; just cook in bacon drippings) and onions. Cook until soft, about 3-5 minutes. Add the carrots and celery and cook another 3-5 minutes. increase the heat to medium, add the mushrooms or other veggies, and saute for 2-3 more minutes.
Make the roux and gravy. Add the garlic, thyme, sage, black pepper, crushed red pepper (if using), remaining butter, and flour and mix into the veggies. Continue to stir and cook the roux for 1-2 minutes. Add the wine and stir to combine. Simmer for 30 seconds. Stir in the chicken stock and bring to a boil. Once boiling, reduce to low and let the sauce thicken for 2-3 minutes.
Preheat oven to 400F + combine, and adjust seasonings and thickness. Taste the sauce and see if it needs more salt, pepper, or herbs. If it's looking a little thick, add a touch of chicken stock. See note 1 if it's still looking a little thin. The sauce should be on the thicker side. Once you have the right thickness, stir in the chicken and pancetta/bacon and remove from heat.
Cut and arrange the puff pastry. With a rolling pin, gently roll out the puff pastry until it’s smooth. Then, using a sharp knife, cut it into 12 even rectangles. Starting with the edge of the skillet, layer the puff pastry pieces over the chicken mixture, allowing ½ inch of the pastry to hang over the edge of the skillet. (This helps to prevent the pastry from falling into the sauce.) Place the remaining puff pastry pieces over the center, covering the filling completely. Don’t worry if there are a few small cracks—that just adds to the charm!
Prep the crust. In a small bowl, whisk together the egg and 1 tablespoon water. Using a pastry brush, brush the egg wash on top of the puff pastry (make sure to get the edges, too!) To finish, sprinkle the puff pastry with thyme and Parmesan cheese.
Bake. Bake for 25 -28 minutes (cover the edges with aluminum foil after 15 minutes to prevent burning) or until the sauce is bubbling and the puff pastry is golden brown.