01 - Preheat the oven to 400°F (200°C).
02 - In a large skillet over medium heat, melt butter. Add onion, carrots, and celery; sauté for 5 to 7 minutes until softened.
03 - Stir in minced garlic and cook for 1 more minute.
04 - Sprinkle flour over vegetables, stirring well to combine. Cook for 2 minutes, stirring frequently.
05 - Gradually add chicken stock and milk, stirring constantly to prevent lumps. Bring to a simmer and cook until thickened, approximately 5 minutes.
06 - Add shredded chicken, peas, parsley, thyme, salt, and pepper. Mix thoroughly and simmer for 2 to 3 minutes. Remove from heat.
07 - Transfer the chicken mixture to a 9-inch deep pie dish.
08 - Unroll the puff pastry and place over the filling. Trim excess pastry and crimp edges to seal. Cut small slits on the top to vent steam.
09 - Brush the pastry surface with beaten egg for a golden finish.
10 - Bake for 25 to 30 minutes until the pastry is golden and crisp.
11 - Allow the pie to rest for 5 minutes before slicing and serving.