Preheat the oven to 400 degrees F. In a large skillet, melt the butter over medium heat. Add the onion, carrots, and celery, and cook for 5 minutes until the vegetables begin to soften.

Stir in the flour and cook for 1 minute to form a roux. Gradually whisk in the chicken broth and milk, then add the salt and black pepper. Continue stirring until the mixture thickens into a sauce, about 3 to 4 minutes.

Remove the skillet from the heat and fold in the cooked chicken and frozen peas until everything is evenly coated.

Transfer the filling to a 9-inch pie dish or similar baking dish. In a small bowl, mix the parsley and rosemary into the pie crust dough if making it from scratch, or press the herbs gently onto the surface of a store-bought crust.

Lay the herb crust over the filling, trim any excess edges, and crimp them against the dish rim. Cut a few small slits in the top to allow steam to escape.

Bake for 25 to 30 minutes until the crust is golden brown and the filling bubbles at the edges. Let the pot pie rest for 5 minutes before serving.