½ cup freshly grated Parmigiano-Reggiano, or as needed
⅓ cup fine dry bread crumbs
1 Tbsp chopped fresh oregano leaves, optional
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. Bake until undersides are golden brown, 10 to 15 minutes, then turn and bake until other sides are lightly browned. Set aside.
Reduce oven temperature to 375 degrees.
Meanwhile, in a large saucepan over medium heat, heat 2 Tbsp 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, breaking up whole tomatoes with your hands.
Cover, reduce heat to low and simmer 15 to 20 minutes.
Add vinegar, basil and salt and pepper to taste. Into 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, the 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 the size of pan and thickness of layers.
Remove from heat and allow to rest for 5 minutes before serving. *Recipe can also be reheated.