Put the 1 cup peeled potatoes and 3/4 cup carrots in a large pot, and cover it with water. Cook over high heat until it boils for several minutes and potatoes and carrots are tender when poked with a fork, but don’t overcook. Drain water and set potatoes and carrots aside.
While the veggies are cooking (or after the veggies cook), in a separate large pot or skillet, combine the 1/2 cup butter (1 stick) and 1/3 cup diced onion over medium heat, while the butter melts, stir often, cooking the onions. When onions are soft, add the 1/2 cup all-purpose flour, 1 teaspoon salt, 1/2 teaspoon dried thyme and 1/4 teaspoons pepper; mix well.
Slowly whisk in the 1 1/2 cups chicken broth or stock and 3/4 cup milk (half and half or heavy cream works too). Stir until the gravy has thickened.
Add the 2 cups cubed cooked chicken, 3/4 cup frozen peas, 1/4 cup frozen corn, and cooked potatoes and carrots to the gravy and stir to combine.
Prepare the pie dish by lining it with an uncooked pastry crust. Pour the chicken and vegetable mixture into the pie shell. Cover with the top pie crust, and let the top layer overlap the bottom crust, pinching seams to seal, then fluting the edge of the seam.
Crack an egg into a small bowl and beat it. Brush the beaten egg over the top, then with a sharp knife, cut some slits in the top of the pie so it can vent steam. Bake in a preheated oven at 425ºF for 35-40 minutes. Let cool for about 10 minutes before digging in. Enjoy!