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Remove the pastry from the refrigerator and unwrap. Allow to soften slightly, then place on a lightly floured work surface and roll the pastry to a thickness of 1/4-inch. Using a small plate or saucer as a guide, cut out 6 (6-inch) rounds. (Scraps may be combined and reformed if you cannot get 6 rounds out of the first batch.) Stack the pastry rounds onto pieces of plastic wrap or parchment paper (with pieces between each round to keep them from sticking together) and refrigerate while you prepare the filling.
Remove the pastry circles from the refrigerator. In a mixing bowl, combine the meat, onion, carrot, potato, salt and pepper and mix until thoroughly combined. Place the pastry circles on a clean work surface and place about 1/2 cup of the filling in the center of 1 side of the pastry. Using the beaten egg, brush the edges of the pastry and then bring the unfilled side over the filled side so that edges meet. Press edges together to seal and then crimp using your fingers or a fork. Repeat with the remaining turnovers and then transfer to a baking sheet. Brush the tops of the turnovers with the remaining egg and then cut several slits into the top of each pastry. Bake for 20 minutes, or until pastry is golden brown around the edges. Reduce the heat to 350 degrees F and continue to bake until the pasties are golden brown. Remove from the oven and allow to cool for 10 to 15 minutes before serving.
*Chef's Note: If you find that the pastry breaks when you try to roll it out, simply gather it together and add a bit more water and knead lightly so that it comes together in a smooth ball. Allow to rest briefly then try again. This pastry is delicate but worth the extra effort.