Preheat your oven to 400 degrees F (200 degrees C) and lightly grease a 9-inch deep-dish pie pan or baking dish.
Melt butter in a large skillet over medium heat. Whisk in the flour and cook for 1 minute to remove the raw flour taste.
Gradually whisk in the chicken broth and heavy cream until the mixture is smooth and begins to thicken.
Stir in the shredded chicken, peas, carrots, potatoes, thyme, and parsley. Simmer for 5 minutes until vegetables are tender.
Pour the warm filling into your prepared baking dish and gently place the puff pastry sheet on top, folding the edges inwards if needed.
Cut a few small slits in the center of the puff pastry to allow steam to escape during baking.
Bake for 20 to 25 minutes until the pastry is golden brown and puffed, and the filling is bubbly.
Remove from the oven and let the pot pie rest for 10 minutes before serving to allow the gravy to thicken.