Preheat the oven to 450 degrees . Halve the eggplants lengthwise and, using a small, sharp knife, score into small crosshatch cubes, being careful not to cut through the skins. Drizzle with 3 tablespoons EVOO and season with salt and pepper. Place cut side down on a baking sheet and roast for 20 minutes.
Meanwhile, in a medium saucepan, heat the remaining 1 tablespoon EVOO, 1 turn of the pan, over medium heat. Add the onion and garlic and cook until softened, 6 to 7 minutes. Stir in the tomatoes and season with salt and pepper. Stir in the basil leaves until wilted. Keep warm over low heat.
Scoop out the eggplant flesh and spread it on the bread. Top with the tomato mixture, parmigiano-reggiano and mozzarella. Place the bread pizzas on the baking sheet and toast in the oven until the cheese melts, about 5 minutes. Cut each loaf half into 4 pieces.
