2. Remove the chicken from the broth with tongs and set aside. When the chicken is cool enough to handle, cut it into small pieces and place it in a large casserole baking dish.
3. Meanwhile, add the carrots, parsnips and potatoes to the broth and heat over high heat. Simmer until they are just tender, about 10 minutes. Transfer them to the baking dish with a slotted spoon. Add the green beans and mushrooms to the simmering broth and cook until they are just tender, about 3 minutes and then transfer them with the slotted spoon to the baking dish.
4. Strain the broth and reserve it in a bowl. Melt the butter in the now empty saucepan over medium heat. Add the flour and stir for 3 or 4 minutes. Whisk in the reserved broth, slowly at first and then more quickly. Bring the sauce mixture to a boil and cook until the mixture thickens enough to coat a spoon. Pour the sauce into the baking dish and season well with salt and pepper.
5. Roll out the dough until it is 1/4 of an inch thick.
(This can be done in advance and stored separately in the refrigerator for 1 day.)
6. Preheat the oven to 400ºF.
7. Place the dough on top of the baking dish and trim it so that it fits inside the dish. Cut a few slits in the dough to allow the steam to escape. Place the pie in the oven and bake until the crust is golden brown and the sauce is bubbling. Let the pot pie stand for 10 minutes before serving.