01 - Preheat the oven to 400°F.
02 - Melt butter in a large skillet over medium heat. Add onion, carrots, and celery; cook for 5 to 7 minutes until softened.
03 - Stir in the minced garlic and cook for an additional minute.
04 - Sprinkle flour over the vegetables, stirring well to combine. Cook for 2 minutes, stirring frequently.
05 - Gradually pour in chicken stock and whole milk while stirring constantly to prevent lumps. Bring to a simmer and cook until thickened, about 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 filling to a 9-inch deep pie dish. Unroll the puff pastry and place it over the filling. Trim excess pastry and crimp edges to seal. Cut a few small slits in the top to allow steam to escape.
08 - Brush the pastry surface with beaten egg to promote browning.
09 - Bake in the preheated oven for 25 to 30 minutes, or until the pastry is golden and crisp.
10 - Allow the pie to rest for 5 minutes before slicing and serving.