Preheat the oven to 375°F.
To prepare the white sauce, melt the butter in a saucepan. Sprinkle in the flour and stir to form a roux.
Gradually whisk in the milk, then add the chicken broth. Season with salt and pepper, then bring to a boil over medium heat.
Reduce the heat to low and simmer until the sauce thickens.
Remove the saucepan from the heat and stir in the chicken and the vegetables.
Let the filling cool for about 15 minutes.
Meanwhile, roll out both pieces of dough to about ⅛ inch thick.
Line a 9-inch pie dish with the first pastry.
Pour the chicken filling into the pastry shell.
Cover with the second pastry and seal the edges together using your fingers or a fork, brushing with a little beaten egg as needed.
Using a knife, cut a small slit in the center of the top crust to allow the steam to escape.
Brush the top crust with the remaining beaten egg yolk.
Bake for about 50 minutes, or until the crust is golden brown.
Serve hot.