Make the crust: Pulse the flour, fine salt and 1/4 teaspoon pepper in a food processor. Add the cheese, butter and shortening; pulse until the butter is in pea-size bits. Add 1/2 cup ice water and pulse until the dough just starts to come together. Turn out onto a sheet of plastic wrap and pat into a disk. Wrap and refrigerate until firm, at least 1 hour and up to 2 days.
Meanwhile, prepare the filling: Discard the chicken skin and shred the meat; set aside. Put the bones in a pot and add the chicken broth. Cover and simmer over medium-low heat, 45 minutes to 1 hour. Strain the broth and set aside.
Preheat the oven to 400 degrees F. Melt the butter in a pot over medium-high heat. Add the onion, thyme, celery, carrots and 1/2 teaspoon kosher salt; cook, stirring, until soft, about 6 minutes. Add the mushrooms and 1/2 teaspoon kosher salt and cook until soft, about 6 more minutes. Add the flour and cook, stirring, 2 minutes. Add the wine; cook until almost evaporated, about 1 minute. Add the broth, cream and 1/2 teaspoon kosher salt. Bring to a boil, then reduce the heat to medium low and simmer until thick, about 3 minutes. Add the parsley, shredded chicken, and salt and pepper to taste. Transfer to a 3-to-4-quart baking dish.
Roll out the dough between 2 sheets of parchment paper until slightly larger than the baking dish. Drape the dough over the filling, press it against the inside of the dish and trim. Brush with cream, then cut a few slits in the top. Put on a baking sheet; bake until golden, about 35 minutes. Let rest 10 minutes before serving.