Preheat the oven to 400 degrees F and arrange three racks evenly spaced.
Lay the eggplant in one layer on three sheet pans and brush both sides with olive oil, using all the oil. Sprinkle with the oregano, crushing it lightly in your hands, then sprinkle with 1 1/2 tablespoons salt and 1 1/2 teaspoons pepper. Bake for 15 minutes. Turn the slices and rotate the pans in the oven and bake for another 10 minutes, until tender. Leave the oven at 400 degrees F.
In a 10 by 14 by 2-inch ceramic baking dish, spread a third of the marinara sauce. Arrange a third of the eggplant on top in one layer. Scatter a third of the basil, a third of the mozzarella, a third of the goat cheese, and a third of the Parmesan on top. Repeat twice, starting with the marinara and ending with the Parmesan, making sure each layer is evenly distributed.
For the topping, place the bread crumbs, garlic, and basil in a food processor and pulse to combine. Add the 1/4 cup olive oil and 1 teaspoon salt and pulse to moisten the crumbs. Sprinkle the mixture evenly over the dish.
Bake for 45 to 50 minutes, until bubbling and golden brown. Allow to sit at room temperature for 10 minutes before serving.