Preheat the oven to 425 degrees.
In a large pot or saucepan over medium heat, melt the butter. Add the onion, carrot and celery, and cook, stirring occasionally, until the vegetables are tender, about 5-7 minutes.
Stir in the flour and cook for 1 minute. Pour in the chicken broth, and stir to combine. Slowly pour in the milk, and add the seasonings. Simmer until thickened. Stir in the cooked chicken.
Place one pie crust in the bottom of a 9 inch pie pan and pour in the filling. Top with other crust and seal the edges.
Whisk together the egg yolk and brush all over the top and sides of the pie crust. Cut slits in the top for steam.
Bake for 30-35 minutes or until golden brown and bubbling. Allow to rest at least 15 minutes before cutting.