01 - Set the oven to 400°F (200°C) to preheat.
02 - Melt butter in a large skillet over medium heat. Add chopped onion, carrots, and celery; cook for 5 to 7 minutes until softened.
03 - Stir in minced garlic and cook for an additional 1 minute.
04 - Sprinkle flour over the vegetables and stir continuously to combine. Cook for 2 minutes to remove the raw flour taste.
05 - Gradually add chicken stock and milk while stirring constantly to avoid lumps. Bring to a simmer and cook until the sauce thickens, approximately 5 minutes.
06 - Add shredded chicken, frozen peas, parsley, thyme, salt, and pepper. Mix thoroughly and simmer for 2 to 3 minutes. Remove from heat.
07 - Transfer the chicken mixture into a 9-inch deep pie dish. Unroll puff pastry over the filling, trim edges, and crimp to seal. Cut small slits on top for steam to escape.
08 - Brush the pastry surface evenly with beaten egg to enhance browning.
09 - Place the pie in the oven and bake for 25 to 30 minutes until the pastry is golden and crisp.
10 - Allow the pie to rest for 5 minutes after baking before slicing and serving.