01 - Preheat oven to 400°F (200°C).
02 - In a large skillet, melt butter over medium heat. Add onions, carrots, and celery. Sauté for 5 to 7 minutes until softened. Add garlic and cook for 1 minute.
03 - Stir in flour and cook for 2 minutes, stirring constantly to form a roux. Slowly whisk in chicken broth, then milk, continuing to whisk until the mixture thickens, about 4 to 5 minutes.
04 - Add salt, pepper, thyme, cooked chicken, and peas. Stir to combine thoroughly. Remove from heat.
05 - Roll out one pastry sheet and line the bottom of a deep 9-inch pie dish or baking dish. Pour in the chicken and vegetable mixture evenly.
06 - Cover with the second pastry sheet. Trim excess dough and crimp edges to seal. Cut small slits in the top to allow steam to escape.
07 - Brush the top crust evenly with beaten egg for a golden finish.
08 - Bake for 35 to 40 minutes, or until the crust is golden brown and the filling is bubbling.
09 - Let rest for 10 minutes before serving to allow the filling to set.