In a dutch oven or pot, melt butter. Add diced onions and carrots and saute 8 minutes over medium heat until soft.
Add sliced mushrooms and minced garlic and saute another 5 minutes until mushrooms are softened.
Add flour and stir constantly for 2 minutes. Add chicken stock, and heavy cream then bring to a simmer and cook 1 minute or until mixture is a thick gravy consistency. Add salt, black pepper, or season to taste. It should be well-seasoned.
Add shredded cooked chicken, frozen peas, and parsley. Stir to combine then remove from heat and cool slightly while you roll out the crusts. Roll 1 chilled pie crust disk into a 12” diameter circle. Carefully transfer it into a deep 9” pie dish. Spoon the pie filling over the bottom crust.
Roll the second disk of pie dough into a 10” diameter circle and place over the pie filling. Fold the excess dough behind the bottom crust then crimp the pie crusts together to seal. Use a sharp paring knife to cut 5 small slits in the top to allow steam to escape. Brush the top of the crust with the beaten egg and sprinkle lightly with coarse salt and pepper.
Bake at 425°F for 30-35 minutes or until top crust is golden brown. If edges are browning too fast, cover with a pie shield or make a shield by cutting a 4” diameter circle from the center of a sheet of foil and placing that over the pie. Once out of the oven, rest for 15 minutes to cool slightly before slicing.