Adjust a rack to the lowest position in the oven. Preheat the oven to 400°F.

Place one pie crust into a 9-inch pie plate and press into the corners. Trim any excess as needed. Line with a sheet of parchment paper large enough to hang over the edges of the plate, and fill with pie weights.

Bake the crust on lowest rack of oven until edges of the crust are very slightly golden and the center is matte, 25-30 minutes.

When done, remove crust from oven and carefully lift off the parchment and pie weights.

Increase the oven temperature to 425°F.

Dice the chicken into bite-size pieces. Set aside.

Dice the onion.

Dice the carrots.

Dice the celery.

Mince the garlic, and remove the thyme leaves from their stems.

Melt butter in a large pot over medium-high heat. Add the onion, carrot, and celery. Saute until softened and translucent, 6-7 minutes.

Reduce heat to medium. Add the garlic, thyme, salt, and black pepper. Saute for a minute longer. Add the frozen peas and corn, and cook until thawed.

Sprinkle the flour into the pot. Stir to coat the vegetables in flour. Cook until the mixture smells toasty, 1-2 minutes.

Slowly stream in the chicken broth, and whisk steadily until the broth begins to boil and thicken.

Whisk in the heavy cream. Allow the mixture to reduce and thicken slightly at a low boil for 1-2 minutes. If it is bubbling rapidly, reduce the heat.

Remove the pot from the heat. Add the chicken, salt, and black pepper. Stir until mixed.

Pour the chicken mixture into the baked crust. Place the second pie crust on top of the pie. Crimp the edges of the pie together with a fork, and trim off any excess as needed. Poke the center of the pie with the fork 2-3 times, so steam can escape.

Beat the egg and water together. Lightly brush the egg wash in a thin layer over the pie. Cover the edges of the pie with a pie guard or strips of aluminum foil.

Bake the pie on lowest rack of oven for 20 minutes.

Reduce the oven temperature to 350°F. Continue baking until the crust is golden and the juices are bubbling slightly at the edges, 25-30 minutes.

Check to see that pie is done. Remove from oven or add time as needed.

Allow pie to sit at room temperature for 20 minutes before slicing and serving. Cover leftovers and refrigerate for up to 3 days.