In a large skillet, melt the butter over medium heat. Add the diced onions, carrots, and celery. Season lightly with salt. Sauté until the vegetables are tender, about 5-7 minutes. The aroma will fill your kitchen, making you wonder if you’ve just stepped into a gourmet restaurant (or your grandmother's kitchen).

Once the vegetables are tender, sprinkle the flour over them. Stir to combine and cook for another minute. This creates what's known as a roux, which will thicken our filling.

Gradually, whisk in the chicken stock while stirring constantly. This prevents any lumps from forming. You’ll notice the mixture starting to thicken already.

Next, add the half-and-half (or your chosen substitute). Stir to combine, bringing the mixture to a gentle simmer. Let it bubble for about 2-3 minutes until it thickens nicely.

Now it’s time to spice things up! Add the shredded chicken, frozen peas, thyme, black pepper, and Dijon mustard. Mix everything well. Let it simmer for an additional 5 minutes, allowing the flavors to meld together. It’s starting to look like a pot pie filling now!

While the filling simmers, preheat your oven to 425°F (220°C). Roll out the refrigerated pie dough on a floured surface. Once your filling is ready, pour it into a pie dish. Carefully place the pie crust over the filling. Trim any excess crust hanging over the edges and seal it by crimping or using a fork.

Cut a few slits in the top of the crust to allow steam to escape. This part is crucial; otherwise, you’ll have a pie that erupts like a volcano. Bake for 25-30 minutes until the crust is golden brown and the filling is bubbly.

Let the pie cool for a few minutes before cutting into it. It will be hard to wait, but your tongue will thank you for it.