Preheat your oven to 400°F (200°C). Place one of the refrigerated pie crust sheets into a 9-inch pie dish, gently pressing it into the bottom and up the sides. Set the pie dish and the second crust aside while you prepare the filling.

In a large skillet or saucepan, heat the unsalted butter and olive oil over medium heat until the butter is melted and slightly foamy.

Add the chopped yellow onion and diced carrots to the skillet. Cook, stirring occasionally, for about 5 to 7 minutes, until the vegetables begin to soften.

Stir in the minced garlic and cook for about 30 seconds, just until fragrant, making sure it does not brown.

Sprinkle the all-purpose flour evenly over the vegetables. Stir well to coat everything, and cook the flour for 1 to 2 minutes, stirring constantly, to remove the raw flour taste.

Gradually pour in the low-sodium chicken broth while stirring, making sure to scrape the bottom of the pan to release any browned bits. Continue stirring until the mixture is smooth and begins to thicken.

Pour in the whole milk and stir again, simmering gently for 3 to 5 minutes, until the sauce is creamy and thick enough to coat the back of a spoon.

Add the shredded or diced cooked chicken, frozen peas, and frozen corn to the skillet. Stir to combine everything evenly in the sauce.

Season the filling with dried thyme, dried parsley, salt, and black pepper. Taste and adjust the seasoning if needed, then remove the skillet from the heat.

Spoon the warm chicken and vegetable filling into the prepared pie crust in the pie dish, spreading it out into an even layer.

Carefully place the second refrigerated pie crust sheet over the top of the filling. Trim any excess dough if necessary, then pinch or crimp the edges of the top and bottom crusts together to seal the pie.

Using a sharp knife, cut 4 to 5 small slits in the center of the top crust to allow steam to escape while baking.

In a small bowl, whisk together the egg and water to make an egg wash. Lightly brush the top crust with the egg wash for a golden, glossy finish.

Place the pie dish on a baking sheet to catch any drips, then transfer it to the preheated oven. Bake for 30 to 35 minutes, or until the crust is golden brown and the filling is bubbling through the slits.

Remove the chicken pot pie from the oven and let it rest for at least 10 to 15 minutes before slicing, so the filling can set slightly. Serve warm.