Preheat the oven to 425°F.

In a large skillet, heat olive oil and butter over medium heat. Add the onion, celery, and garlic; sauté until softened, about 5 minutes.

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.