Make the dough and, while it is rising, make the pizza sauce: In a medium saucepan, heat the olive oil over medium-high heat. Add the garlic and cook, stirring, for 30 seconds. Add the salt and red pepper, and cook, stirring, for 30 seconds. Add the tomatoes, wine and sugar, and bring to a boil. Lower the heat and simmer, stirring occasionally, until very thick, 40 to 45 minutes. The sauce needs to be very thick, with no visible liquid, otherwise your pizzas will be wet. Remove from the heat and stir in the fresh herbs; let cool completely before using.

Preheat a grill on the high setting to about 500 degrees F.

Oil 2 seasoned 12-inch round deep-dish pizza pans (we used cast iron skillets) generously with the vegetable oil. Press 1 piece of dough into each pan, using your fingers to press the dough in an even layer all along the bottom and coming as far up the sides of the pan as possible. Let rest for 5 minutes.

Layer the sliced mozzarella cheese all over the bottom of the pies. Top each with half of the salami, mortadella, ham and olive salad.  Ladle the sauce evenly over each pizza and top with the Provolone, Parmesan, and Pecorino. Using your fingertips, gently fold the top of the dough over the edges of the filling all around the pizza and sprinkle the exposed dough edges with sesame seeds.

Place the pizza pans on the rack of the grill so that they’re only getting indirect heat. Close the grill cover and reduce the grill settings so that the temperature hovers between 450 to 475 degrees F. Cook until the top of the pizzas are golden, the cheese is bubbly, and the crust is golden brown, usually 22 to 24 minutes. Remove from the grill and set aside to rest for at least 10 minutes before serving. Slice and serve hot.