Combine the flour, sugar and salt in the bowl of a food processor fitted with the steel blade. (Alternatively, you can prepare the pie crust by hand in a large bowl and use a pastry blender, your fingers, or two forks to work in the butter.) Pulse a few times to mix.

Add the cold butter cubes and pulse 8-10 times, until the largest butter pieces are the size of small peas.

Add 6 tablespoons of ice-cold water. Pulse just until the dough just starts to come together, adding one more tablespoon of cold water if the dough seems too dry. The dough should still look crumbly at this point, but it should hold together if you press it between your fingers.

Transfer the dough to a lightly floured work surface. Use your hands to bring the dough together into a ball and then cut the dough in half. Shape each half of the dough into a round disk. Wrap both disks of dough in plastic wrap and place them in the refrigerator while you prepare the pot pie filling.

Preheat oven to 425° F.

In a large Dutch oven or other heavy pot, melt the butter over medium heat.

Add the chopped onion, carrots and celery to the pot. Cook, stirring occasionally, until softened, about 5 minutes.

Stir in the minced garlic and flour and cook, stirring, for 1 minute.

Stir in the broth and milk, followed by the dried thyme, salt and pepper. Cook, stirring often, until the mixture begins to bubble and thickens, about 2 minutes.

Remove the pot from the heat and stir in the shredded chicken and frozen peas.

Prepare the egg wash by combining the egg and 1 tablespoon milk or water in a small bowl and whisking until well combined. Set aside.

Take one of the disks of pie dough out of the refrigerator and, on a lightly floured work surface, roll it out to a 12-inch circle. Transfer it to a deep dish pie dish by gently rolling the dough circle around your rolling pin. Lightly fit the dough into the pie dish. Do not press it down or the crust may stick to the pie dish after baking.

Similarly, roll the second disk of pie dough out into a 12-inch circle.

Pour the chicken pot pie filling into the bottom pie crust in the pie dish and spread it into an even layer. Place the second rolled out pie crust on top of the pie. Trim the dough overhang to be no more than ½-inch past the rim of the pie dish. Fold the edges of the pie dough under and then use your fingers to crimp as desired for a decorative edge.

Cut a few slits in the top pie crust to let steam escape as the pot pie bakes. Then lightly brush the egg wash all over the pie crust. (This will help it to brown nicely as it bakes.)

Bake for 30-35 minutes, until crust is golden brown. Let rest for 10-15 minutes and then serve.