Preheat the oven to 400F.

Melt the butter in a large Dutch oven over medium heat, add the onion, carrots and celery, and cook 6 to 8 minutes until softened.

Sprinkle the flour over the vegetables, stir, and cook 1 minute to coat.

Slowly whisk in the chicken broth, then the milk, stirring constantly until smooth.

Bring to a simmer and cook 5 to 7 minutes, stirring, until the sauce thickens enough to coat the back of a spoon.

Stir in the shredded chicken, peas, thyme, 1 teaspoon salt and 1/2 teaspoon pepper.

Pour the filling into a 9x13 inch baking dish.

Drape the pie crust over the filling, trim the overhang to 1/2 inch, crimp the edges, and cut 4 small slits in the center for steam to escape.

Brush the crust with the beaten egg.

Bake 30 to 35 minutes until the crust is deep golden brown and the filling bubbles up through the slits.

Cool 15 minutes before serving so the filling sets enough to hold its shape when scooped.