Chicken and Mushroom Pot Pies

1. 1In the bowl of a food processor, stir together the flour and salt. Sprinkle the butter over the top and pulse for a few seconds, or just until the butter is slightly broken up into the flour but still in visible pieces. Evenly sprinkle the water over the flour mixture, then process just until the mixture starts to come together. If the dough looks too dry, add 1 or 2 additional tablespoons of water.
2. 2Dump the dough onto a work surface, press it together, then divide it in half. Press each half into a disk, then place the disks in a large zipper-top plastic bag. Refrigerate the dough for at least 30 minutes or as long as 1 day.

To make the filling:

1. 1In a saucepan over medium heat, poach the chicken in gently simmering water until opaque, about 15 minutes. Remove with a slotted spoon and place on a cutting board. Let cool slightly, then shred the meat, discarding the bones and any bits of gristle and fat. You should have 3 cups of meat. Set aside.
2. 2Heat the oven to 400°F. In a large, wide saucepan, cook the carrots and leeks in 1 tablespoon of the butter and the olive oil with a sprinkle of salt, stirring occasionally, until the carrots are crisp-tender, about 15 minutes. Scrape into a bowl. Melt another 1 tablespoon butter then add the mushrooms. Cook, stirring, until tender, about 10 minutes. Add to the carrot mixture. Set aside.
3. 3Melt the remaining 5 tablespoons of butter in the saucepan. Sprinkle the flour over the top and whisk in. Cook for a few minutes until bubbly. Slowly add the chicken stock, whisking constantly until smooth. Whisk in the milk.
4. 4Add the reserved chicken, carrots, leeks, mushrooms, and peas and thyme to the sauce. Stir to combine. Taste for salt and add more if necessary. Season with black pepper. Set aside.

To assemble and bake:

1. 1Have ready 4 fairly deep, 1-1/2-cup ramekins or baking dishes. On a lightly floured work surface, roll out the dough to a large rectangle about 1/8 inch thick. Cut out 4 circles that are 1/2 to 1 inch larger than the circumference of your ramekins or baking dishes. You will have leftover dough; press the scraps together and save them for another use in a plastic freezer bag (they can be frozen for up to 3 months).
2. 2Divide the filling among the ramekins and top with the pie dough. Tuck the edges of the pie dough underneath and fit the dough just inside the ramekin. Place the ramekins on a baking sheet. Cut a few slashes in each crust for steam to escape during baking. Brush the egg and water mixture on the dough. Bake until bubbly and the pastry is brown and cooked through, about 40 minutes.