Preheat the oven to 400°F.
Melt the butter in a 14-inch cast iron or ovenproof skillet. Add the onion, carrots, and thyme and cook, stirring often, until softened, about 2 minutes.
Sprinkle the flour over the vegetables and stir. Cook, stirring over medium heat for 1 minute.
Add the broth, stirring constantly. Add the cream, chicken, peas, and smoked paprika. Bring to a simmer, stirring often, and cook until sauce has thickened, 2 to 3 minutes. Season with salt and pepper to taste.
Roll out the pie crust so that it’s about 1 inch larger than the pan you’re using. Place the crust on top of the pot pie mixture, and press crust into the sides of the pan. Brush the crust with the egg wash and cut vents in the top of the crust.
Bake until very golden and bubbly and the crust is done, 30 to 40 minutes. Let cool for a little bit before serving.