In a Dutch oven or a large pot, melt six tablespoons of butter over medium heat. Add the diced onions and carrots, and sauté for approximately eight minutes, or until they are soft and translucent.

Incorporate the sliced mushrooms and minced garlic into the pot. Continue to sauté for an additional five minutes, until the mushrooms have softened.

Stir in one-third cup of flour, ensuring it is well combined with the vegetables. Continue stirring for two minutes. Gradually add the chicken stock and half a cup of heavy cream, bringing the mixture to a simmer. Cook for one minute or until it reaches a thick, gravy-like consistency. Season with two teaspoons of salt and one-quarter teaspoon of black pepper, adjusting to taste.

Add the shredded cooked chicken, frozen peas, and one-quarter cup of parsley to the mixture. Stir thoroughly to combine, then remove from heat. Allow the mixture to cool slightly while you prepare the pie crusts. Roll out one disk of the pie dough into a 12-inch diameter circle and carefully place it into a 9-inch pie dish. Spoon the pie filling over the bottom crust.

Roll out the second disk of dough into a 10-inch diameter circle. Place it over the pie filling. Fold any excess dough underneath the bottom crust and crimp the edges to seal the pie. Using a sharp paring knife, cut five small slits in the top crust to allow steam to escape during baking. Brush the top crust with the beaten egg and lightly sprinkle with coarse salt and pepper.

Preheat the oven to 425°F (220°C). Bake the pie for 30-35 minutes, or until the crust is golden brown. If the edges brown too quickly, cover them with a pie shield or a foil ring. Once baked, allow the pie to rest for 15 minutes before slicing and serving. This resting period allows the filling to set and makes slicing easier.