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1. Simmer the potato in a small pot of salted water until just tender, 8 to 10 minutes, then drain and set aside.
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2. Heat the oil in a large skillet over medium heat. When it shimmers, add the onion and cook, stirring occasionally, until translucent, 2 to 3 minutes. Add the carrot, celery, and raw chicken, season with salt and pepper to taste, and raise the heat to medium-high. Cook until the vegetables are softened and the chicken is just about cooked, 3 to 5 minutes.
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3. Sprinkle the flour over everything, stir, and cook for 1 minute. Add the broth, thyme, and reserved potatoes, and simmer for 2 minutes more. Transfer the mixture to a bowl and refrigerate until cool, 15 to 30 minutes. Remove the pie crusts from the refrigerator so they can soften slightly.
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4. When the filling is cool, preheat oven to 425u00b0F and set the racks in the upper and lower thirds of oven. Grease or line two baking sheets.
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5. On a lightly floured work surface, roll out 1 pie crust into a rough rectangle (don't roll it too thin or the handpies will fall apart). Use a sharp knife or pizza cutter to square off the rounded edges, then cut the dough into 4 equal pieces.
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6. Place about 1/4 cup of filling slightly off-center on 1 piece of dough, then fold the dough over to form a rectangle. Crimp the edges together with a fork, and prick the top a few times for steam to escape. Carefully transfer pie to the prepared baking sheets. Repeat with remaining pie crust and filling.
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7. Brush each pie lightly with the egg wash, then bake until golden brown, 20 to 25 minutes.
