Preheat oven to 425°F. In a skillet, melt butter over medium heat. Add diced onion, carrots, celery, salt, and pepper. Sauté until vegetables soften, about 5-7 minutes. Stir in dried thyme.

Add the chicken pieces to the skillet and cook until they are lightly browned on all sides, about 5 minutes. Sprinkle in the flour and stir well, cooking for another minute to remove the raw flour taste.

Gradually pour in chicken broth and half-and-half, stirring constantly. Cook until thickened, about 3-4 minutes. Stir in frozen peas and remove from heat; cool slightly.

On a lightly floured surface, roll out one sheet of the refrigerated pie crust and fit it into a 9-inch pie dish. Spoon the cooled chicken and vegetable filling into the crust.

Roll out the second crust and lay it over the top, pressing the edges together to seal. Trim any excess dough and cut a few slits in the top for steam to escape. Brush the crust lightly with the beaten egg.

Place the pie in the preheated oven and bake for 30-35 minutes, or until the crust is golden brown and the filling is bubbly. Allow the pot pie to cool for about 10 minutes before serving.