How to Make It
Butter a 13-by-9-by-2-inch baking dish. In a saucepan, combine the turkey stock, potatoes, thyme and a large pinch of salt and bring to a boil over high heat. Reduce the heat to low, cover and simmer until the potatoes are tender, about 10 minutes.
Meanwhile, in a large skillet, melt 1 tablespoon of the butter in 1 tablespoon of the oil. Add the white mushrooms, season with salt and pepper and sauté over moderately high heat until golden, about 5 minutes. Transfer to a bowl. Repeat the process with 1 more tablespoon each of the butter and oil and the shiitakes. Add to the white mushrooms.
Melt the remaining 1 tablespoon of butter in the remaining 1/2 tablespoon of oil. Add the onion and prosciutto and sauté over moderately high heat until the onion softens, about 5 minutes. Remove from the heat and add the mushrooms.
Strain the turkey stock into the skillet. Add the flour mixture and bring to a boil over moderate heat, stirring constantly. Add the turkey, potatoes and parsley and season with salt and pepper. Spread the mixture in the baking dish and let cool.
Preheat the oven to 400°. If necessary, roll out the puff pastry on a lightly floured surface to a 14-by-10-inch rectangle. Moisten the rim of the baking dish with water and cover the filling with the dough, pressing it firmly against the baking dish rim; tuck the edges under. Make a few slits in the pastry and brush with the beaten egg. Bake for 45 minutes to 1 hour, or until the top is golden. Let the potpie stand for 10 minutes before serving.