Preheat the oven to 400°F.
Heat olive oil and butter in a 2¾ quart Dutch oven until nice and hot and the butter has begun to foam. Add the leek, carrot, celery, garlic, thyme, and herbes de Provence and sauté for 10 minutes, until vegetables are softened and have begun to brown.
Add the flour and cook for 1 to 2 minutes, stirring constantly.
Pour in your veggie stock and stir until the sauce has thickened and has a creamy texture. This should take about 8 to 10 minutes.
Remove from the heat and fold in your shredded chicken, frozen peas, and chopped parsley. Remove the thyme sprig, taste for seasoning, and add more salt and pepper as needed. Set aside to cool (the pan needs to be fully cooled before you add the pasty to the top or it will melt).
Meanwhile, lightly flour a clean work surface and unfold puff pastry. Roll the sheet out and cut a 12-inch circle (making sure the circle is large enough that it will hang over the edge of the Dutch oven by at least 1 inch).
In a small bowl, whisk together the egg and 2 tablespoons of water.
Brush the egg wash around the outside edges of the cooled Dutch oven and carefully place the dough on top of the Dutch oven and press the overhanging edges around the outside of the pot to stick.
Using a knife, cut a few slits in the top of the puff pastry, and brush the entire pastry shell with the egg wash which will give the pie a lovely golden brown color. Place the pot pie on a baking sheet with parchment paper and bake in the oven for 30 minutes.