Chill the bowl and blade of a food processor for 1 hour before starting.
For the dough: To the food processor, add the flour and salt and pulse to combine. Add the cold butter and the cheese and pulse 3 or 4 times or until chickpea-sized pieces of butter remain. With the machine running, drizzle in the ice water until the dough just starts to come together. Turn the dough out onto a piece of plastic wrap and form into a flat disk. Wrap tightly and refrigerate for at least an hour and up to 2 days.
Position an oven rack on the lower third of the oven. Preheat the oven to 400 degrees F.
For the filling: In a medium bowl, combine the apples, sugar, lemon juice, flour, salt and cinnamon. Toss well to coat evenly.
Divide the dough in half. Roll out one of the dough pieces to an approximately 11-inch round and line the bottom of a 9-inch pie plate. Fill the pie crust with the apple mixture. Roll out the remaining dough into another 11-inch round and place on top. Trim the edges so there is a 1/2-inch overhang around the plate. Tuck the overhang under itself and press down to seal the edge. Using 2 knuckles and the tip of your finger, crimp the edges to make a decorative edge. Cut an X in the center of the top pie crust and open the corners to make a small hole in the center of the pie. Brush the pie evenly with the egg and sprinkle with the Parmesan cheese.
Place on a rimmed baking sheet and bake until the crust is deep golden brown and the mixture inside is bubbling, about 45 minutes. Remove to a wire rack and cool for at least 2 hours before serving.