Remove all the meat from the chicken and discard the skin and bones; hand-shred the meat. Reserve.
In a large saucepan, add butter and onions and sauté over medium heat until onions become translucent.
Add flour and cook over low heat, stirring constantly, for about 3 minutes. Add chicken stock and continue cooking, stirring constantly, until the sauce becomes thick. Add heavy cream, salt, and pepper. Add chicken, carrots, peas, and parsley. Mix until incorporated. Set aside and allow cooling.
For the dough, place flour, sugar, and salt in the bowl of a stand mixer fitted with the paddle attachment and mix for about 30 seconds. With the mixer at the lowest speed, add butter and shortening to the flour mixture. Mix until it resembles large pieces of breadcrumbs.
Slowly, add iced water in a thin stream. Add water to the dough until it barely holds together. Do not overmix. Dump the dough out onto a floured surface and knead into a ball. Try not to overwork the dough too much. Wrap the dough in plastic and allow to rest in the refrigerator for at least 30 minutes.
Preheat oven to 375°F.
Put the chicken filling onto 4 individual ovenproof dishes (you may have extra filling, you can reserve up to 3 days in the refrigerator for some other time). Divide the dough in four and roll each piece enough to cover the dish and that you have an extra inch to the side. Brush the sides of the dishes with egg wash and put the rolled dough on top. Press the dough to the sides of the dish. Brush the top of the dough with egg wash and sprinkle shredded parmesan cheese on top. Make 3 slits on top to allow steam to escape.
Place the pot pies in a baking sheet and bake for 45 to 50 minutes or until golden brown.