Make the pie crust: combine the flour, sugar, and salt in a food processor with the dough attachment.

Add the cubed butter and process until butter is about pea size. Add the buttermilk and process until it forms a ball. If it is too dry add a couple tablespoons of water. Once it holds together, divide the dough in two and form into disks. Wrap each disk in plastic wrap and refrigerate while you make the filling.

To make the filling, heat the butter over medium heat in a large skillet.

Add the onions, carrots, celery, and garlic and cook until tender, stirring occasionally.

Add the flour stirring continuously for 30 seconds. Add the salt, black pepper, thyme, parsley, chicken broth, and heavy cream.

Stir until sauce is smooth and slightly thickened.

Simmer over medium low heat for 10 minutes. Stir in the shredded chicken and frozen peas. Remove from heat and set aside while you prepare 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.

Transfer dough to a 9-inch deep dish pie pan.Using your fingers press the dough to the sides and trim the dough with a knife.

Fill pie with the filling.

Roll out the second disk of dough and carefully cover the pie. Trim the excess dough leaving a slight over hang (about an 1/2"). Turn the excess dough under itself and seal the edges by crimping with a fork or using your favorite method.

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. You will need some type of cover for the edges or they might burn. I simply cut our a circle from a piece of tin foil.

Bake for 45 minutes, or until crust is golden brown.

Cool for 10 minutes, cut into slices and serve.