Keep the pie crust refrigerated until the pot pie filling is ready for the oven. Preheat the oven to 425F.
In an oven-safe skillet (I use a 12" cast iron skillet), over medium-high heat, melt the butter. Once melted, add the veggies. Saute the veggies for 10 minutes, until they begin to sweat and the onions are translucent. Stir in the shredded chicken. Sprinkle the thyme, salt, pepper, and flour evenly over the chicken and veggie mixture. Stir to evenly coat the chicken and veggie mixture. Stir in the broth and milk. Cover the skillet and allow the mixture to cook, over medium heat, until slightly thickened (about 5-7 minutes). Add the peas to the filling, and stir. Remove the skillet from the heat.
Roll out the pie crust so it will cover the skillet. Drape the pie crust over the skillet. You're welcome to crimp the sides of the pie crust, but I don't. This is optional: I like to place the skillet on a parchment-lined, or foil-lined, baking sheet to prevent any spillage (and avoid dirtying the baking sheet).
In a small bowl, whisk together the egg and water. Brush the top of the pie crust with the egg wash. This is completely optional, but it adds a rich golden color to the final crust, which I love. Pierce the center of the crust with a knife to allow the steam to escape while baking. You can see my two slits in the center of my crust via the photos.
Bake the pot pie for 30 minutes, until the crust is golden brown.