Preheat the oven to 400°F. Butter the bottom and sides of a 9-inch pie plate. Line the bottom of the pie plate with one of the pie crusts and set aside.

In a large pot, add chicken and cover with water. Add a little salt and pepper. Bring to a boil and cook for 20 minutes, until the meat is easily shredded.

While the chicken is cooking, melt the butter in a heavy bottomed skillet. Sauté the onion and celery for 5 minutes, until softened. Add the garlic and continue to sauté for 1 minute.

Sprinkle the flour over the sautéed vegetables and stir to combine to create the roux for your gravy. Cook for about 2 to 3 minutes, making sure to continuously stir, until the flour slightly browns and becomes fragrant.

While still stirring, slowly add in the broth and milk until everything is well combined. Bring to a simmer and cook until the gravy starts to thicken, about 4 minutes. Once the gravy is starting to thicken, add the peas and carrots, thyme, and shredded chicken. Stir and season to taste with salt and pepper.

Add the chicken filling to the prepared pie dish and lay the other pie crust on top. Pinch the pie crust edges together, sealing the crust. Brush the top of the pie crust and the edges with the beaten egg. Use a fork to poke a few holes in the center of the pie crust to let the steam out as the pot pie bakes.

Bake for 30 to 40 minutes, until the pie crust is golden brown. If the edges of the crust are browning too quickly, cover just the edges with aluminum foil.

Let the pot pie rest 10 minutes before serving to allow the gravy filling to set up. Then slice and serve warm.