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.
Preheat the oven to 375 degrees Fahrenheit. Butter or line a baking sheet.
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.
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.