Make the dough: Put 2 1/4 cups warm (100 degrees F) water in a large measuring cup; stir in the yeast to dissolve, then stir in the sugar and olive oil. Let stand until bubbly, 3 to 5 minutes. Combine 5 cups flour and 2 teaspoons salt in a stand mixer fitted with the paddle attachment. Add the yeast mixture and mix on medium-low speed, adding a little more flour or water, if needed, until a rough, sticky ball of dough forms, about 1 minute. Let rest 5 minutes, then mix on medium low until no longer sticky, about 1 more minute. Transfer the dough to an oiled surface and knead with oiled hands until smooth, about 10 times. Let rest 5 minutes, then knead briefly again and transfer to an oiled bowl. Cover with plastic wrap; refrigerate at least 8 hours or overnight. Divide the dough into 4 balls, dust with flour and cover loosely with plastic wrap. Let sit at room temperature 2 hours before making the pizzas.
Make the sauce: Crush the tomatoes in a bowl using your hands. Stir in the olive oil, garlic, 1 1/2 teaspoons salt and the oregano. Cover and let stand 2 hours. Position a pizza stone or inverted baking sheet on the lowest oven rack; preheat to 500 degrees F for at least 30 minutes. Stretch each ball of dough into a 12-to-14-inch round on individual sheets of parchment paper. Let rest 15 minutes. Remove and discard the garlic from the sauce. Spread a thin layer of sauce on each pizza, leaving a border. Top with one-quarter each of the mozzarella, parmesan, cherry peppers, soppressata and radicchio; drizzle with olive oil. Slide 1 pizza (on the parchment) onto a pizza peel or another inverted baking sheet, then slide onto the hot stone in the oven. Bake until the crust browns, 12 to 15 minutes. Top with oregano. Repeat to make 3 more pizzas. Photograph by Con Poulos