The versatile and delicious topping for this pizza is a sautéed shortcut version of ratatouille, which is sometimes called ciambotto in Italy. It cooks in only 15 minutes and is so flavorful and has such a great melting texture that no sauce is needed for the pizza.

• Put a baking stone on the lowest rack in your oven. (Move the top rack up or out of the oven for easiest maneuvering). Heat the oven to 500 degrees F.
• In a large (12-inch) nonstick skillet, heat 3 Tbs. of the olive oil over medium heat. Add the eggplant, peppers, and 1/4 tsp. salt, and turn the heat to medium high. Cook, stirring occasionally at first and more frequently as the vegetables begin to brown, until the vegetables are softened and all of them have some browning (some will be very brown), 10 minutes. Reduce the heat to medium and add the cherry tomatoes, the remaining Tbs. of olive oil, and 1/4 tsp. salt. Cook, stirring and scraping the pan occasionally, until the tomatoes have started to collapse (some will brown) and the mixture is well-integrated, 5 minutes. Remove the pan from the heat and drizzle with the balsamic glaze or syrup, if using. Season to taste with salt. Transfer the mixture to a plate or tray and divide it loosely into thirds.
• Combine the oil and garlic in a small bowl and stir to combine.
• Arrange the garlic oil, topping, cheeses, and salt near your work area.
• Sprinkle a pizza peel or the back of a large cookie sheet with cornmeal. Lightly flour your work surface, your hands, and a rolling pin. Gently press one dough ball out with your fingers and hands until it forms a flat cake on your work surface. Using the rolling pin, roll the dough out to an 8-inch circle. If the dough is bouncy, let it rest between rolls.
• Transfer the rolled dough to the pizza peel or baking sheet.
• Spoon about 1 scant Tbs. of the seasoned oil (including the garlic) onto the dough and brush or use the back of a spoon to spread it all over the dough, leaving a 3/4-inch border around the edge. Spoon one portion (a third) of the topping on the pizza and nestle 6 to 9 pieces of the mozzarella in it. Sprinkle with a few teaspoons of Parmigiano-Reggiano and drizzle with a bit more of the garlic oil. Sprinkle with a pinch of salt.
• Transfer the pizza to the oven and cook until the crust is golden brown around the edges and crisp and golden on the bottom (check with tongs), 8 to 9 minutes. Use the peel (or tongs and a baking sheet) to transfer the pizza to a wooden cutting board. Top with a third of the fresh basil. Slice and serve.
• Repeat with the remaining dough and topping to make two more pizzas.