Preheat your oven to 400°F (200°C). Lightly grease a 9-inch deep-dish pie plate or a large cast-iron skillet.
In a large saucepan or skillet, melt 1/4 cup of butter over medium heat. Whisk in 1/4 cup of flour and cook for 1 minute to remove the raw flour taste.
Gradually whisk the chicken broth and heavy cream into the flour mixture until smooth. Simmer for about 5 minutes, or until the sauce is thickened and coats the back of a spoon.
Stir the cooked chicken, peas and carrots, parsley, salt, and black pepper into the sauce. Pour the warm filling into the prepared baking dish.
In a large mixing bowl, combine 2 cups of flour, baking powder, and salt for the crust. Cut in the cold, cubed butter using a pastry blender or two forks until the mixture resembles coarse crumbs.
Pour in the cold buttermilk and stir gently with a wooden spoon until a shaggy dough forms. Turn the dough out onto a lightly floured surface and gently pat or roll it out into a 9-inch circle to match your baking dish.
Carefully place the buttermilk biscuit dough over the warm chicken filling. Cut 3-4 small slits in the top to allow steam to escape during baking.
Bake in the preheated oven for 25 to 30 minutes, or until the crust is golden brown and the filling is bubbling around the edges.
Remove from the oven and let the pot pie rest for 10 to 15 minutes before serving to allow the filling to set slightly.