Cook the chicken: Cut chicken into chunks and season lightly with salt and pepper. In a skillet, cook until no longer pink. Remove and shred or dice.
Prepare the filling base: In the same pan, melt butter. Add onion, celery, and garlic. Sauté until soft and fragrant.
Make the sauce: Sprinkle in flour and stir well. Slowly whisk in chicken broth and milk until smooth. Let it thicken for a few minutes.
Add vegetables and chicken: Stir in peas, carrots, thyme, and the cooked chicken. Taste and adjust seasoning with salt and pepper.
Assemble the pie: Roll out one pie crust in a 9-inch pie dish. Spoon in the filling evenly. Place the second crust on top. Seal the edges and cut small slits for steam to escape.
Bake: Brush the crust with the beaten egg. Bake at 400°F (200°C) for 35–40 minutes, or until the crust is golden brown.
Rest and serve: Let the pie cool for 10 minutes before cutting. This helps the filling set.