Preheat your oven to 400°F. Press one pie crust into a 9-inch deep dish pie pan and set aside.
In a large skillet over medium heat, melt the butter. Add onion, carrots, and celery. Cook for 5-7 minutes until vegetables are softened.
Sprinkle flour over the vegetables and stir constantly for 1-2 minutes to cook out the raw flour taste.
Slowly pour in the chicken broth and milk, stirring constantly. Bring to a simmer and cook until mixture thickens, about 5 minutes.
Stir in the turkey, peas, thyme, salt, and pepper. Remove from heat and let cool slightly.
Pour the turkey mixture into the prepared pie crust. Top with the second pie crust, crimping the edges to seal. Cut 4-5 slits in the top to allow steam to escape.
Brush the top with beaten egg for a golden finish. Bake for 30-35 minutes until the crust is golden brown and filling is bubbly.
Let cool for 10 minutes before slicing. Serve warm and enjoy the compliments!