Preheat the oven to 180°F. Oil a parchment-lined baking sheet.
Place the eggplant slices on the pan and brush on both sides with 2 tablespoons of the olive oil. Sprinkle on the oregano, salt, and pepper. Bake for 10 minutes, turning once. They will then be supple.
While the eggplant is in the oven, make a simple tomato sauce by whirring the tomatoes briefly in a food processor. In a medium skillet over medium heat, sauté the onion for 2 to 3 minutes in the remaining 1 tablespoon of olive oil, then add the garlic and sauté for another minute. Stir in the tomatoes and cook the mixture briefly, just to blend flavors, about 2 minutes.
Remove the eggplant from the oven. Lower the oven temperature to 350°F. On each eggplant piece, place a slice of prosciutto and a slice of mozzarella. Roll the pieces from the small end forward, and secure the neat little bundle with a toothpick or by tying a chive around it.
Slather the bottom of a 9 x 13-inch baking dish with some of the tomato sauce, and arrange the involtini seam side down. Over each bundle spread some more tomato sauce and a scattering of the Parmigiano. Warm well in the oven, about 15 minutes.