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Special equipment: a 4-inch round cookie cutter
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For the dough: Whisk the egg with 3 tablespoons cold water in a small bowl.
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Pulse the flour, 1/3 of the butter, Parmesan, vinegar, 1/2 teaspoon salt and 1/4 teaspoon pepper in a food processor until the mixture looks like fine meal.
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Add the remaining butter and pulse until the mixture is in pea-size pieces.
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Add the egg and pulse until the dough begins to come together.
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Divide the dough in half; place each half on a sheet of plastic wrap and pat into a disk.
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Wrap the disks tightly and refrigerate until firm, at least 1 hour up to overnight, or freeze for up to 2 months.
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For the filling: Meanwhile, add the oil to a large Dutch oven or pot over medium-high heat.
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Add the onion, celery and thyme and cook, stirring, until soft, about 5 minutes.
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Add the mushrooms, 1 teaspoon salt and a few grinds of pepper and cook, stirring, until soft, about 4 minutes.
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Add the flour and cook, stirring, 2 minutes.
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Add the broth, peas and carrots and sour cream.
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Bring to a boil, then reduce the heat to medium-low and simmer until thick, about 3 minutes.
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Stir in the chicken and parsley and add salt and pepper to taste.
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Allow the mixture to cool completely.
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(The filling can be made a day ahead.)
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Preheat the oven to 375 degrees F. Line two baking sheets with parchment.
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Roll 1 disk of dough between two sheets of parchment to about 1/8-inch thick.
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Using a 4-inch round cookie cutter, cut into nine rounds.
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Gather the scraps, reroll and cut out 3 more rounds.
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Cover the rounds with a clean, damp towel to prevent the dough from drying out.
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Repeat with the remaining disk of dough, for a total of 24 rounds.
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(If the dough gets too warm while you're working with it, chill it until it's easier to work with.)
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Place 1 round of dough on a clean, lightly-floured work surface and roll out to about 4 1/2 inches.
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Place 1/3 cup of filling in the center and brush the edges with beaten egg.
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Roll another round of dough out to 4 1/2 inches and drape over the filling; seal the edges and crimp with a fork.
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Brush the entire pot pie with more egg, cut a few slits in the top, sprinkle with some pepper and place on a prepared baking sheet.
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Repeat with the remaining dough and filling.
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Bake the pot pies until they're golden brown and the filling begins to bubble, 20 to 24 minutes.
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Let cool slightly before serving.
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Alternatively, freeze the unbaked pot pies for up to 1 month, then bake, from frozen, in the oven, preheated to 375 degrees F, until they're golden brown and the filling begins to bubble, 20 to 24 minutes.
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Let cool slightly and serve.
