Preheat oven to 450 degrees. Brush both sides of eggplant slices with oil, and place in a single layer on two or more baking sheets lined with parchment paper. Bake until undersides are golden brown, 10 to 15 minutes, then turn and bake until other sides are lightly browned. (The last time I made this, I was in a hurry and did not turn the slices over and the dish turned out fine.) Set aside. Reduce oven temperature to 375 degrees.
Meanwhile, in a large saucepan over medium heat, heat 2 tablespoons olive oil and add onion. Sauté until soft, about 10 minutes. Add garlic and dried oregano and sauté another 30 seconds. Add tomatoes and their juices. Cover, reduce heat to low, and simmer 15 to 20 minutes.
Add vinegar, basil (if using), and salt and pepper to taste. Into a lasagna pan or a 9-by-9-inch, 10-by-5-inch, or 10-by-6-inch baking pan, spoon a small amount of tomato sauce, then add a thin scattering of parmigiano, then a single layer of eggplant. Repeat until all ingredients are used, ending with a little sauce and a sprinkling of parmigiano.
In a small bowl, combine bread crumbs and oregano, if using, with just enough olive oil to moisten. Sprinkle on top. If desired, recipe can be made to this point and refrigerated. Bring to room temperature before baking.
Bake until eggplant mixture is bubbly and center is hot, 30 to 45 minutes depending on size of pan and thickness of layers. Remove from heat and allow to rest for 5 minutes before serving. Can also be reheated.