First, make the pie crust. Combine the flour, sugar, and salt in a large bowl. Add the cubed butter and toss to coat. Dump the mixture out onto a clean surface and use a rolling pin to roll the butter into thin sheets, combining it with the flour. Use a bench scraper to scrape the rolling pin and to bring the mixture back into a pile as necessary. Continue until all of the butter is incorporated into the flour. Mixture will be very flaky. Return mixture to the bowl and place in the freezer for 15 minutes to chill the butter.
Remove from freezer and add the buttermilk. Use a spoon and then your hands to stir the mixture until it comes together into a ball. If mixture is too dry, add the water a tablespoon at a time. Divide the dough in two and flatten into disks. Wrap each disk in plastic wrap and chill in the fridge while you make the filling.
To make the filling, heat the butter over medium-high heat in a large skillet. Add the onions, carrots, celery, and garlic and cook until tender, stirring occasionally.Whisk in the flour, salt, black pepper, thyme, parsley, chicken broth, and heavy cream. Whisk until there are no flour lumps and then simmer over medium-low heat for 10 minutes or until sauce has thickened. Stir in the shredded chicken or turkey and frozen peas. Remove from heat and set aside while you roll out the pie dough.
Preheat oven to 400°F.
Remove the pie dough from the refrigerator. On a lightly floured surface, use a rolling pin to roll out the dough into a 12-inch circle. Dough should be about 1/4 inch thick. Transfer dough to a 9-inch pie pan. Pat with your fingers, making sure it is smooth. Trim the extra overhang of dough with a knife and discard. Fill pie with filling. Roll out the second disk of dough and carefully cover the pie. Trim the extra overhang off the sides. Seal the edges by crimping with a fork or your fingers. With sharp knife, slice a few small slits in the center of the top crust. Using a pastry brush, brush crust and edges with beaten egg.
Bake for 45 minutes, or until crust is golden brown. I use a pie crust shield to protect the edges from browning too much too soon. Cool for 10 minutes, cut into slices and serve.
Note-we love making this recipe when we have leftover rotisserie chicken or turkey. This pot pie freezes well. Cool completely and freeze for up to 1 month. You can freeze the whole pie or slices.