First, melt the butter in a large skillet over medium heat. Add the sliced leeks and sauté them for about 5 to 7 minutes until they become soft and translucent. It's important not to brown them, as this keeps their flavor delicate and sweet.
Next, sprinkle the flour over the softened leeks and stir it in well. Cook for another minute to remove any raw flour taste. Gradually whisk in the chicken broth, ensuring no lumps form. As the mixture heats, it will start to thicken into a smooth sauce.
Lower the heat and pour in the heavy cream, stirring until it's fully combined. Add the Dijon mustard, fresh thyme, salt, and pepper. Let this sauce simmer gently for about 5 minutes, allowing all the flavors to meld together into a rich, aromatic base.
Finally, stir the diced cooked chicken into the creamy sauce. Mix everything together until the chicken is evenly coated. Remove the skillet from the heat and allow the filling to cool slightly while you get the pastry ready. This prevents the bottom crust from becoming soggy.
Begin by preheating your oven to 400°F (200°C). While the oven heats, take one sheet of puff pastry and gently press it into a 9-inch pie dish. Make sure it lines the bottom and sides evenly, then trim any excess pastry from the edges.
Pour the slightly cooled chicken and leek filling into the pastry-lined dish, spreading it out into an even layer. Now, place the second sheet of puff pastry over the top. Trim the edges and press them together firmly to seal the pie.
For a decorative touch, you can crimp the edges with a fork. Use a sharp knife to cut a few small slits in the top crust. These vents are crucial, as they allow steam to escape during baking, which keeps the pastry crisp.
Lastly, brush the top of the pie with the beaten egg. This egg wash is what will give your pie a beautiful, glossy, and golden-brown finish once it's baked. Place the pie on a baking sheet to catch any potential drips.
Place the pie in your preheated oven and bake for 25 to 30 minutes. You'll know it's ready when the crust is puffed up, flaky, and a deep golden brown. The filling should be bubbling hot, which you might see through the vents.
Once baked, carefully remove the pie from the oven. It's important to let it cool for at least 10 minutes before you slice and serve it. This waiting period allows the creamy filling to set, ensuring you get clean, beautiful slices.