Melt butter in a large saucepan over medium heat and cook leek and mushrooms for 8 minutes until softened and their liquid has mostly evaporated.
Stir in flour for 1 minute, then gradually whisk in broth and milk. Simmer for 5 minutes until thick enough to coat a spoon, then stir in chicken, peas and carrots, thyme, salt and pepper.
Transfer filling to shallow containers and refrigerate within 2 hours. Chill completely, then divide among four individual metal pot-pie pans that each hold about 2 cups.
Roll pastry on a lightly floured surface into a 12-inch square and cut into four 6-inch lids large enough to cover the pans with a small overhang. Lay over cold filling, press to the rims and cut one steam slit in each.
Freeze before baking: wrap each pie tightly, label and freeze for up to 2 months. Reserve the egg and water for baking day rather than freezing them with the pies.
Thaw pies in the refrigerator for 24 hours. Heat the oven to 400°F and set unwrapped pies on a rimmed metal sheet pan. Beat egg with water and brush the pastry lightly.
Bake for 30 to 40 minutes until pastry is golden and the center of the filling reaches 165°F, loosely covering the pastry with foil if it browns first. Rest for 10 minutes and serve one pie per portion.
Reheat a cooked refrigerated pie at 350°F for 25 to 35 minutes until its filling reaches 165°F, covering the pastry loosely if needed.