Make the dough first. In a large bowl, whisk together flour, sugar, and salt. Using a pastry blender or your fingers, cut in the cold butter until the mixture resembles coarse crumbs with some pea-sized pieces remaining.
Add the eggs. Beat 2 eggs in a separate small bowl, then add to the flour mixture. Stir until just combined.
Add ice water gradually, 1 tablespoon at a time, until the dough comes together when pinched. It should be just moist enough to hold together, not wet.
Divide the dough into two portions: 2/3 for the bottom crust and 1/3 for the top. Flatten each portion into a disk, wrap in plastic wrap, and refrigerate for at least 1 hour.
While the dough chills, make the filling. In a large bowl, beat the ricotta with 4 eggs until well combined. Add all the diced cheeses, meats, Pecorino Romano, parsley, black pepper, and red pepper flakes if using. Mix thoroughly.
Preheat your oven to 375°F (190°C) and grease a 9-inch springform pan.
Roll out the larger dough disk on a floured surface to about a 14-inch circle, approximately 1/8-inch thick. Carefully transfer to the springform pan, pressing it into the bottom and up the sides, leaving some overhang.
Blind bake the crust. Line the dough with parchment paper and fill with pie weights or dried beans. Bake for about 15 minutes until the edges just start to turn golden.
Remove the weights and parchment, then pour the filling into the par-baked crust, spreading it evenly.
Roll out the remaining dough to a 10-inch circle and place it over the filling. Trim both the top and bottom crust edges, leaving about 1/2 inch overhang, then fold and crimp them together to seal.
Beat the remaining egg and brush it over the top crust. Cut a few small slits in the center for steam to escape.
Bake for about 1 hour until the crust is golden brown and a knife inserted in the center comes out clean. If the crust starts to brown too quickly, cover loosely with foil.
Let it cool in the pan for about 30 minutes, then remove the springform ring and let cool completely before slicing.