Preheat oven to 400°F (200°C).
Melt butter in a 12-inch cast iron skillet over medium-high heat.
Add chicken cubes and cook until browned, about 5-7 minutes. Remove chicken and set aside.
Add onion, carrots, and celery to the skillet. Sauté until softened, about 5 minutes.
Stir in minced garlic and cook for 1 minute.
Sprinkle flour over the vegetables and stir for 2 minutes to create a roux.
Gradually whisk in chicken broth and heavy cream, simmering until the sauce has thickened.
Fold in the cooked chicken, frozen peas, salt, pepper, and thyme.
Lay the puff pastry sheet over the skillet, trimming or tucking the edges as needed.
Cut three small slits in the center of the pastry to allow steam to escape.
Brush the surface of the pastry with the beaten egg.
Transfer the skillet to the oven and bake for 20-25 minutes. The crust should be golden brown and the filling bubbling.
Allow to rest for 10 minutes before serving.