1. In a small bowl, whisk melted butter and flour. Set aside.
2. In a large frying pan over medium heat, saute onion, carrots, celery, and garlic in olive oil. Season with salt and pepper as needed. Once the onions are clear, add the mushrooms, thyme, and parsley. Once the mushrooms have cooked down, add the roux and pour the bone broth (or stock) a little as you go. Add the half and half and keep stirring until thickened. Finally, add the chicken and string beans. Toss and season again if needed with salt and pepper.
3. Remove from the heat and proceed with the pastry shell.

Classic Flaky Crust:

1. Preheat oven to 400 degrees F.
2. In a food processor, combine flour, salt, butter, and shortening. Pulse until the mixture becomes crumbly. Add ¼ cup of ice water and pulse again until a soft dough begins to form. You may need to add more water if the dough is still sandy. Add one tablespoon at a time as needed.
3. Form the dough into a disk and wrap in clingfilm. Refrigerate for at least 30 minutes or up to overnight.
4. Turn the dough out onto a well-floured surface and divide it half. Roll out one half into ¼-inch thickness and fit into a pie plate.
5. Roll out the other half of pie dough into ¼-inch thickness. Fill the pie shell with the chicken and vegetable mixture, then cover with the other half of pie dough. Crimp the edges and slice a vent at the top to release the steam as the pie bakes.
6. In a small bowl, whisk together 1 egg and 1 tablespoon of water. Brush the entire pie with the egg wash. Bake the pie for 15 minutes at 400 degrees F. then reduce the heat to 350 degrees F. until browned and bubbly, about 30-35 minutes.