Place the warm water and sugar in a small bowl, add the yeast and allow to soften for 15 minutes, until it has dissolved and is foaming. In a separate small bowl, combine 2/3 cup olive oil, 1 tablespoon of salt and the milk. In a large bowl, combine the flour and cornmeal and add the yeast mixture and the olive oil mixture, stirring a few times with a wooden spoon to form a rough amalgam. Turn the mixture out onto a lightly floured board and knead for 6 to 8 minutes, until it is smooth and resilient. Lightly oil a large bowl, place the dough in the bowl and cover tightly with plastic wrap. Set aside in a warm place to rise for 35 to 40 minutes, until it has doubled in bulk. Divide the dough into 2 equal portions and, with a well-seasoned rolling pin, quickly roll the first half out to 1/4-inch thickness. Using a 5-inch biscuit cutter, cut rounds from the dough and place on a sheet tray. You may combine the scraps and re-roll the dough once. Repeat the process with the other half of the dough, until all is used up. This should yield 10 disks. Cover the disks lightly with parchment or a clean cloth and allow to rest for 10 minutes.

Divide the mozzarella into 10 equal portions and place a portion on 1 half of each disk, half-moon style. Add pecorino, prosciutto and tomatoes to each mound of mozzarella. Season with pepper and fold the disks in half to form half-moons. Press the edges with your finger to seal them well. Cover again with parchment or a clean cloth.

Preheat the oven to 150 degrees F.

In a large, heavy-bottomed saucepan that will hold liquid at a depth of at least 5 inches, heat the olive oil until almost smoking. Add 3 or 4 of the pastries at a time and cook until browned, turning gently with tongs or a wooden spoon to ensure even browning on both sides. As the panzarotti finish cooking, remove them to a tray lined with paper towels, and hold them in the oven until ready to serve.