01 - Preheat oven to 400°F (200°C).
02 - Heat olive oil in a large skillet over medium heat. Add diced onion and cook until translucent, about 3 minutes.
03 - Add minced garlic, diced carrots, celery, and sliced mushrooms. Cook for 5 to 7 minutes until softened.
04 - Stir in shredded chicken, frozen peas, chopped parsley, dried thyme, salt, and black pepper. Remove from heat and set aside.
05 - Melt unsalted butter in a saucepan over medium heat. Whisk in all-purpose flour and cook for 1 minute.
06 - Gradually whisk in chicken broth and whole milk. Stir constantly until sauce thickens, approximately 4 minutes.
07 - Pour the thickened sauce over the chicken and vegetable mixture. Stir thoroughly to combine.
08 - Transfer filling into a 9-inch (23 cm) pie dish. Roll out pastry to cover the top, trim excess, and crimp edges securely.
09 - Cut small slits in the pastry to allow steam to escape. Brush surface with beaten egg for a golden finish.
10 - Bake in preheated oven for 30 to 35 minutes until crust is golden brown and crisp.
11 - Allow pie to rest for 10 minutes after baking before slicing and serving.