01 - Preheat the oven to 400°F (200°C).
02 - In a large skillet over medium heat, melt the butter. Add the onion, carrots, and celery. Cook for 5–7 minutes until softened.
03 - Stir in the garlic and cook for 1 minute more until fragrant.
04 - Sprinkle the flour over the vegetables and stir well to combine. Cook for 2 minutes, stirring frequently to cook out the raw flour taste.
05 - Gradually pour in the chicken stock and milk, stirring constantly to prevent lumps. Bring to a simmer and cook until the sauce has thickened, about 5 minutes.
06 - Add the shredded chicken, peas, parsley, thyme, salt, and pepper. Mix well and simmer for 2–3 minutes. Remove from heat.
07 - Transfer the chicken mixture to a deep 9-inch pie dish.
08 - Unroll the puff pastry and lay it over the filling. Trim excess pastry and crimp the edges to seal securely. Cut a few small slits in the top to allow steam to escape.
09 - Brush the pastry surface evenly with the beaten egg.
10 - Bake for 25–30 minutes, or until the pastry is golden brown and crisp.
11 - Allow the pie to rest for 5 minutes before serving to let the filling set.