Preheat oven to 450ºF.
Brush the eggplant slices, both sides, with the olive oil. Placing each slice in a single layer on baking sheet(s), careful not to overlap too much.
Bake for 10 to 15 minutes, or until undersides are golden. Turn over to bake other side until lightly browned, about 8-10 minutes. Set aside.
Reduce oven temperature to 375ºF.
Meanwhile, over medium heat, heat 2 tablespoons olive oil, until warm but not smoking.
Add chopped onion and sauté until translucent. About 10 minutes.
Add sliced garlic and dried oregano. Sauté for about 30 seconds.
Add tomatoes and their juices, breaking them up with your hands.
Cover and reduce heat to low.
Simmer for 15 to 20 minutes.
Tear the basil leaves into small pieces and add to the sauce.
Stir in the vinegar.
Add salt and pepper to taste.
Prepare either a 9-by-9-inch, 10-by-5-inch or 10-by-6-inch baking pan, by spreading a small amount of tomato sauce on the bottom.
In layers, sprinkle a small amount of Parmigiano, followed by a single layer of eggplant.
Repeat the layers, with the last ones being the sauce and a sprinkling of Parmigiano.
Combine bread crumbs and oregano, in a small bowl.
Add a little olive oil, just enough to moisten the breadcrumbs. Sprinkle on top.
Bake for 35-45 minutes, or until mixture is bubbling and heated throughout.
Let rest for 5-10 minutes before serving.