In a large saucepan, cook the carrot, celery, leek and garlic in the butter over medium heat for about 5 minutes or until tender. Season with salt and pepper. Add the flour and cook for 1 minute while stirring. Add the broth and bring to a boil while stirring. Cover and simmer gently for about 5 minutes. Add the chicken, peas and cheese. Blend well and adjust the seasoning. Allow to cool.
With the rack in the lowest position, preheat the oven to 190 °C (375 °F).
In a small bowl, whisk together the egg yolk and milk. Set aside.
Line a 23-cm (9-inch) pie plate with a pie crust. Spoon in the chicken and vegetable mixture. Cover with the second pie crust. Make an incision in the centre. Tightly seal the edges by crimping it with a fork or your fingers. Brush with the egg wash. Bake for 50 minutes or until the crust is golden brown.