Line a baking sheet with aluminum foil and set aside. Dust your countertop with flour, and lay the puff pastry flat out on the floured surface. dust the top of the puff pastry with flour as well, and roll it out to 1/8" thick. If you use Dufour, you won't need to roll it much at all. Using the soup ramekins you intend to use as a guide, cut out rounds from the pastry ~3/4" wider on all sides than the rim of the ramekins. One package of Dufour puff pastry should yield 5-6 rounds. Place the dough rounds on the lined baking sheet and place in the fridge. Preheat the oven to 425°F.
In a large sauté pan, melt the butter over medium heat. Add the diced shallot, and cook for 1-2 minutes until fragrant, stirring occasionally. Add the onion, celery, carrots, and garlic, stirring occasionally for 8-10 minutes. The onions should be getting translucent and the carrots should be cooked, but still firm.
Add the 1/3 cup flour and stir constantly for 2 minutes as the filling thickens. Add the chicken broth, salt and pepper and stir, making sure to scrape up any bits from the bottom of the pan. Bring the filling to a boil, then reduce to a simmer and cook for a few minutes until the filling has thickened to your desired texture. Remove the pan from the heat and stir in the heavy cream, thyme, shredded chicken, and frozen peas. Once everything is mixed, taste now and adjust salt/pepper as necessary.
Using a ladle or a large spoon, add the filling to the ramekins or soup bowls, making sure not to overfill; I recommend no more than 3/4 full. Remove the pastry rounds from the fridge. Brush the rims and outer edges of the ramekins with the beaten egg, and then place the dough rounds over the top of the ramekins. Press the dough around the edges to stick the dough to the egg wash. Place all the ramekins on the lined baking sheet, and then with the remaining egg wash, brush the tops of all the dough. With a sharp knife, cut a small 1/2" hole in the top of the dough to allow steam to escape.
Bake the pot pies for 15-20 minutes, until the top is golden brown. The filling is already cooked before it goes in the oven, so you're just looking for a good top crust, no need to overbake. Let the pot pies cool for 10 minutes (they will be hot!) before eating.
Enjoy!
