Preheat the oven to 400?F. Grease a 9 x 9 x 2-inch baking pan.
Heat the butter in a large sautÈ pan over medium-high heat. Add the onions and celery, season with salt and pepper, and cook, stirring, for 2 minutes. Stir in the flour and cook, stirring, for 3 to 4 minutes to make a blond roux. Stir in the chicken stock and bring the liquid to a boil. Reduce the heat to medium-low and simmer until the sauce begins to thicken, 4 to 6 minutes. Stir in the half-and-half and continue to cook for another 4 minutes. Season with salt and pepper. Stir in the potatoes, carrots, peas, chicken, and parsley, season with salt and pepper, and mix well.
Line a baking pan with one of the rolled-out piecrusts. Pour the filling into the prepared pan. Place the second crust on top of the filling. Carefully tuck the overlapping crusts into the pan, forming a thick edge. Crimp the edges, cut vents in the top crust, and place on a baking sheet. Bake until the crust is golden brown and crusty, 25 to 30 minutes. Let cool for 5 minutes before slicing to serve.
Yield: 6 servings