Preheat the oven to 400 degrees F. Melt 2 tablespoons of butter in a large skillet over medium heat. Add the chicken pieces and cook until lightly browned on all sides. Remove the chicken from the skillet and set it aside.

Add the remaining 2 tablespoons of butter to the same skillet. Stir in the carrots, celery, and onion. Cook for 5 minutes until the vegetables begin to soften.

Sprinkle the flour over the vegetables and stir for 1 minute. Slowly pour in the chicken broth while stirring to avoid lumps. Add the heavy cream, thyme, salt, and pepper. Bring the mixture to a simmer and cook until it thickens slightly.

Return the chicken to the skillet along with the frozen peas. Stir everything together and remove from heat. Transfer the filling to a baking dish.

Lay the puff pastry sheet over the top of the filling. Trim any excess dough and crimp the edges against the dish. Cut a few small slits in the center for steam to escape. Brush the pastry evenly with the beaten egg.

Bake for 25 to 30 minutes until the crust is puffed and deep golden brown. Let the pot pie rest for 5 minutes before serving.