Add the mushrooms and cook until they are tender, about 3-4 minutes.
Sprinkle the flour over the mixture and stir to combine. Cook for 1-2 minutes to eliminate the raw flour taste.
Gradually whisk in the chicken broth and milk, stirring constantly until the mixture thickens, about 5 minutes.
Stir in the shredded chicken, peas and carrots, thyme, salt, and pepper. Remove from heat and let cool slightly.
Roll out one pie crust and fit it into a 9-inch pie dish. Pour the chicken and mushroom filling into the crust.
Roll out the second pie crust and place it over the filling. Seal the edges by crimping with a fork or your fingers. Cut a few slits in the top crust to allow steam to escape.
Brush the top crust with the beaten egg for a golden finish.
Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown. Let cool for 10 minutes before serving.
Notes
For a creamier filling, substitute half of the milk with heavy cream.
Incorporate fresh herbs like parsley or rosemary for added flavor.