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Preheat the oven to 500 degrees F.
1. Place the eggplant in a large bowl, drizzle about 2 tablespoons of the olive oil on top, and gently toss to give the eggplant a light coating of oil. Arrange the slices as close as possible to each other on a lightly oiled sheet pan without overlapping the slices. Using the same bowl and more oil, repeat the yellow squash zucchini. Place the yellow squash and zucchini next to the eggplant, or, if there isn't room, on a second oiled sheet pan.
2. Add all but 1 tablespoon of the remaining oil to the bowl. Dip the onion slices in the bowl, keeping the slices whole, if possible, and place them side by side on another sheet pan. Dip the tomato slices in the bowl, and place them on the same pan or, if there isn't room to avoid layering, on a separate pan.
3. Season all the vegetables with salt and pepper, and roast for 13 to 15 minutes, in two batches if you don't have enough room in your oven. The yellow squash and zucchini should be browned on their bottoms, the onions should be starting to brown, and the tomatoes starting to shrink and tenderize.
4. Remove the vegetables and reduce the oven temperature to 350 degrees F.
5. Brush an 8- or 9-inch square pan that's about 2 inches deep with oil. Gently layer about half the eggplant on the bottom of the pan, and layer the yellow squash on top of the eggplant. Press down firmly to pack the vegetables together. Layer the zucchini on top, then the remaining eggplant, and then the onions, pressing down after each. With a spatula, gently cover the onions with the tomatoes. Top with the basil leaves.
6. Combine the garlic, bread crumbs, cheese, any oil remaining in the bowl, and additional salt and pepper, and mix with your hands. Sprinkle the topping evenly over the lasagna. Bake for about 1 hour. Check during that hour and if the lasagna seems to have a lot of liquid coming from the vegetables, drain some of it. The lasagna is done when it starts to brown on top and pulls back from the pan a bit. Let it cool about 20 to 30 minutes before slicing or serving. Cut in quarters for entrees, in eighths for side dishes.
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