Prepare the Filling Base: In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onion, carrots, and celery. Cook, stirring occasionally, until the vegetables are tender-crisp, about 8-10 minutes.

Add Seasonings and Flour: Stir in the dried thyme and black pepper. Cook for another minute until fragrant. Sprinkle the flour over the vegetables and stir well to coat. Cook for 2-3 minutes, stirring constantly, to cook out the raw flour taste.

Create the Sauce: Gradually whisk in the chicken broth, a little at a time, ensuring no lumps form. Once all the broth is incorporated and the mixture is smooth, slowly whisk in the half-and-half (or whole milk). Bring the mixture to a gentle simmer, stirring constantly, until the sauce thickens to a gravy-like consistency, about 5-7 minutes.

Combine Ingredients: Remove the pot from the heat. Stir in the cooked chicken, frozen green peas, and optional frozen corn. Season with salt, tasting and adjusting as needed. Let the filling cool slightly while you prepare the crusts.

Assemble the Pie: Preheat your oven to 400°F (200°C). On a lightly floured surface, unroll one of the pie crusts. Carefully place it into a 9-inch pie plate, pressing it gently into the bottom and up the sides. Pour the cooled chicken filling into the pie crust, spreading it evenly. Unroll the second pie crust and place it over the filling. Trim any excess dough from the edges, leaving about a ½-inch overhang. Crimp the edges of the top and bottom crusts together to seal. Cut several slits in the top crust to allow steam to escape during baking.

Egg Wash and Bake: In a small bowl, whisk the egg with 1 tablespoon of water to create an egg wash. Brush the top crust evenly with the egg wash; this will give the pie a beautiful golden sheen. Place the pie on a baking sheet and bake for 40-50 minutes, or until the crust is deeply golden brown and the filling is bubbling hot.

Rest and Serve: Once baked, remove the pie from the oven and let it rest for at least 10-15 minutes before slicing and serving.