Preheat oven to 375°.

In a large pan, melt butter over medium heat. Add potatoes, green beans, corn, carrot, onion, and celery; cook until vegetables are just soft, about 7 minutes.

In a bowl whisk together broth, half-and-half, and flour. Add mixture to skillet. Stir in parsley, 1¼ teaspoons salt, pepper, and thyme. Bring to a boil over medium-high heat, stirring frequently. Stir in chicken.

On a lightly floured surface, roll piecrust to a 12-inch circle. Place in pie dish.

Add half of vegetable mixture to pie crust.

Cover with second pie crust. Cut 6 slits in top of dough to release steam. Place pie on a baking sheet.

In a small bowl, whisk together egg and 1 teaspoon water. Brush dough with egg mixture, and sprinkle with remaining ¼ teaspoon salt.

Bake until crust is golden brown and filling is bubbly, about 40 minutes. Let stand for 15 minutes before serving.