Combine flour, salt, butter, and shortening in the bowl of food processor. Pulse until mixture resembles coarse cornmeal. Stir in eggs and pulse until a soft ball forms. If dough is too dry, add ice water a tablespoon at a time until dough is moist. Roll dough in ball. Pinch off about a third; roll each into ball and wrap in plastic wrap. Refrigerate for 1 hour.
In large mixing bowl, beat ricotta, Romano or parmigiana, eggs, pepper, and parsley until blended. Stir in chopped meats and mozzarella.
Heat oven to 375 degrees. On a lightly floured surface, roll out the large dough so that it fits in a 9 x 3-inch springform pan. Fit dough in pan, pushing against sides and bottom to smooth out folds. Add filling to prepared pan. Roll out remaining dough into 10-inch circle. Cut dough in 1/2-inch strips and lay across top of filling, about 1/2-inch apart. Rotate pan and lay dough crisscross across pan. Pinch dough edges together. Whisk together water and egg; brush on top to seal.
Bake pie until golden brown and filling is set and heated through, about 1 hour. Cool for 10 minutes before removing from pan. If serving warm, let pie set for about 20 minutes before slicing. Cool to room temperature and refrigerate.