Preheat oven to 425 degrees F (220 degrees C).

In a Dutch oven over medium heat, melt the butter. Add the celery, onion, and carrots and sauté for 12 minutes.

Season with the thyme, black pepper, and salt.

Sprinkle in the flour and cook until browned, about 1 minute.

Gradually whisk in the chicken broth and rosemary until smooth. Bring to a light simmer and cook for 2 minutes.

Stir in the half-and-half, peas, and parsley.

Add the chicken and cook for an additional 2 minutes, or until heated through.

Remove from the heat and pour into your pie plate that has been lined with pie crust.

Top with the remaining pie crust and trim the edges. Cut several slits in the top of the crust to allow steam to escape.

In a small bowl, whisk together the egg and 1 tablespoon of water. Brush the egg wash over the top of the pie crust.

Bake for 30 minutes, or until the crust is golden brown and the filling is bubbling.

Let cool for 10 minutes before serving.