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Place the chicken in a pot and add water to barely cover. Bring to a boil, reduce the heat, and simmer for 50 minutes. Skim and discard any foam as it rises on the surface. Remove the chicken to cool continue to boil the broth to reduce and concentrate to about 1 quart. Remove the meat from the chicken and shred.
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To make the filling, melt the butter in a medium-sized hot skillet and add the onion, celery, and carrot. Saute over medium heat for 2 to 3 minutes. Stir in the salt and flour and cook for 1 minute more. Add the chicken broth and stir until thickened, about 2 minutes. Stir in 1 heaping cup shredded chicken and the Parmesan cheese. Cool in the fridge.
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Preheat the oven to 375 degrees Fahrenheit. Butter or line a baking sheet.
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To form the pocket pies, work with half of a disk of dough at a time, rolling it out on a floured surface. Using an overturned bowl (about 5 inches across), cut out circles about 3 at a time from each piece of dough. After cutting out all your circles, gather all dough scraps, reroll, and cut out a final time. Place 1/4 cup filling on one side of a dough circle. Wet the edges of the dough with the water. Fold the dough over to form a half circle. Pinch the edge of the dough together. Crimp the edge with a fork. Repeat the process until all the filling is used. The pocket pies can be frozen at this point.
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Place the pocket pies on the prepared baking sheet and chill for a few minutes. Prick each pie on top twice with a fork. When ready to bake, beat the egg with 1 tablespoon water. Brush the egg wash over each pocket pie. Bake for 20 to 25 minutes, until golden brown. Let rest for 5 minutes before serving. The pies can be cooled and frozen to reheat in the microwave.
