Steps:

• 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.

Steps:

• Place the chicken into a pot with the water, add salt, and simmer over medium heat until meat is falling off of the bones, about 40 minutes. Remove from heat, and cool. Remove chicken from broth, reserving broth. Remove the meat from the chicken, cutting large pieces into chunks. Discard skin and bones.
• Meanwhile, place the potatoes into a large pot, and fill with enough water to cover. Bring to a boil over, and cook 8 minutes. Add the celery, and cook 2 minutes more. Drain, and place into a large bowl.
• Bring a second pot of water to a boil over medium-high heat. Add the carrots, and cook 5 minutes. Add the onion and cook 3 minutes more. Remove from heat and drain.
• Preheat oven to 350 degrees F (175 degrees C). Lightly grease a 9x13 baking dish.
• Melt 4 tablespoons of butter in a skillet over medium heat. Stir in the carrots and onion, and cook until tender, about 15 minutes. Mix with the potatoes and celery. Stir in the chicken.
• To make the sauce, melt 7 tablespoons of butter in a pot over medium heat. Whisk 6 tablespoons of flour into the butter, and cook until light brown and paste-like, about 3 minutes. Slowly whisk in the cream and 2 cups of the reserved chicken broth. Continue whisking until the mixture thickens, about 5 minutes. Stir in the parsley and green onions. Season to taste with salt and pepper. Pour the cream sauce over the vegetables and chicken mixture, tossing to coat evenly. Spoon the mixture into the prepared baking dish.
• To make the Cheddar crust, cut 4 tablespoons butter into 2 cups of flour until the mixture is pea-sized. Stir in the Cheddar cheese and sprinkle the ice water over the mixture, gathering the dough into a soft ball. Knead gently until smooth and elastic. Roll dough out on a lightly floured surface until large enough to cover the entire baking dish. Place over the chicken mixture in the baking dish, tucking in edges to fit. Brush with the beaten egg yolk. Pierce top in a few places with a fork or knife to vent steam.
• Bake in preheated oven until top is golden; 30 to 35 minutes.