In a large skillet, heat the olive oil over medium heat. Add the chopped onions, carrots, celery, and garlic. Sauté for about 5-7 minutes or until the vegetables start to soften.
Stir in the diced potatoes, and cook for another 3-4 minutes.
Sprinkle the flour over the vegetable mixture and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the chicken broth and milk, ensuring there are no lumps. Continue to stir until the mixture thickens (about 5-7 minutes).
Add the shredded chicken, peas, thyme, salt, and pepper. Mix thoroughly and remove from heat.
Unroll one pie crust and fit it into a 9-inch pie dish. Pour the chicken filling into the crust, spreading it evenly.
Cover the filling with the second pie crust. Trim any excess and crimp the edges to seal the pie. 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 and the filling is bubbling.
Remove from the oven and let it sit for 10 minutes before slicing.
Notes
Serve warm, garnished with fresh parsley, and enjoy with a side of mixed greens for a beautiful meal.