Preheat your oven to 200°C (390°F) and prepare your pie dish by setting it aside.
In a large skillet, melt a tablespoon of butter over medium heat until it begins to smell nutty and golden.
Add the chopped onion to the skillet and sauté for about 5 minutes, stirring occasionally, until it becomes translucent and fragrant.
Stir in the minced garlic and cook for another minute until it releases a warm aroma.
Add the shredded chicken to the skillet and mix well, warming it through for about 2 minutes.
Pour in the chicken broth and add the frozen peas and carrots, stirring to combine. Let everything simmer for 5 minutes until the vegetables are tender and the mixture is slightly thickened.
Stir in the cream, dried thyme, and chopped parsley, then remove the skillet from heat and set aside.
Roll out the pie dough on a floured surface to fit your pie dish, then transfer it carefully into the dish, pressing gently to fit the bottom and sides.
Pour the warm filling into the prepared crust, spreading it evenly.
Cover the filling with the second sheet of rolled-out dough, trim the edges, and seal by pinching or crimping. Cut small vents on top to allow steam to escape.
Brush the top crust with the beaten egg to give it a beautiful, shiny golden finish.
Bake the pie for 30–35 minutes until the crust is golden and crisp. Let it rest for about 10 minutes before slicing and serving hot.