Step 1

Heat the oven to 425°F with racks in the upper- and lower-middle positions. Brush 2 rimmed baking sheets with 2 tablespoons oil each.

Step 2

Remove half of the skin from each eggplant. To do this, using a vegetable peeler and working from top to bottom, peel off strips of skin about 1 inch apart. Cut the eggplants crosswise into ¼-inch-thick rounds, then arrange in a single layer on the prepared baking sheets, slightly overlapping.

Step 3

Brush the eggplant with 2 tablespoons of the remaining oil, then sprinkle lightly with salt and pepper. Roast until the slices are spottily browned, the edges are slightly crisped and the moisture has cooked off, 35 to 40 minutes; halfway through, use a wide metal spatula to flip the slices, then switch the positions of the baking sheets. Remove from the oven; leave the oven on.

Step 4

Meanwhile, in a large saucepan over medium-low, heat the remaining 1 tablespoon oil until shimmering. Add the onion and ½ teaspoon salt; cover and cook, stirring occasionally, until the onion is translucent, 4 to 5 minutes. Stir in the tomatoes, butter, ½ teaspoon pepper and ½ cup water. Bring to a simmer, then reduce to medium-low and cook, stirring occasionally, until the sauce is slightly thickened, 15 to 20 minutes. Remove from the heat and cool for 15 minutes. Taste and season with salt and pepper.

Step 5

Measure 1 cup of the tomato sauce into a 9-by-13-inch baking dish and spread to cover the bottom. Arrange one-third of the eggplant on top of the tomato sauce, overlapping the slices if needed, then sprinkle evenly with 1 cup provolone, ¼ cup Parmesan and about half of the basil. Spoon on another ¾ cup tomato sauce and spread evenly. Layer on half of the remaining eggplant, followed by the remaining provolone, ¼ cup of the remaining Parmesan and the remaining basil. Spoon on another ¾ cup sauce and layer on the remaining eggplant. Cover with the remaining sauce, spreading it evenly, then sprinkle with the remaining Parmesan.

Step 6

Bake on the lower rack until the edges are bubbling and the cheese is melted, 15 to 20 minutes. Cool for about 10 minutes before serving.