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1
Start by making the tomato sauce. In a large saucepan, heat 1/4 cup olive oil over medium heat. Add the onion (and a pinch of salt) and cook for a few minutes until it softens; then add the garlic and wait for both to turn a light golden brown.
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2
Add the thyme and carrot and cook five minutes more; then add the tomatoes and juice, a pinch of salt, bring to a boil-stirring often-and then lower to a simmer. Cook like this for 30 minutes, stirring occasionally, until the sauce looks thick and tastes terrific (adjust with more salt, as necessary). Set aside.
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3
Preheat the oven to 450 degrees F.
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4
Oil a baking sheet with olive oil.
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5
Slice each eggplant into 6 pieces about 1 to 1/2-inches thick; place as many slices as you can on to the baking sheet (you'll have to do this in batches), drizzle them with a little more olive oil, salt and pepper. Pop into the hot oven and bake until they're deep brown on top, about 15 minutes. Remove the eggplant slices to a plate to cool and roast the remaining eggplant.
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6
Lower the oven temperature to 350 degrees F. In a large baking pan, begin your layering. Start with the largest pieces of eggplant on the bottom, then a layer of tomato sauce, basil (if using), one slice of mozzarella for each eggplant slice, and a sprinkling of Parmesan. Top with a layer of smaller eggplant slices, more sauce, more basil, more mozzarella and more Parmesan. If you have more eggplant and cheese to work with, you can do another layer, but I stopped here. Finish by topping the whole baking pan with the breadcrumbs.
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7
Place in the oven and bake until the bread crumbs are golden on top and the cheese is melted, about 20 minutes. Serve it up hot!
