Using some extra-virgin olive oil, oil a baking sheet.
Slice each eggplant into 6 pieces about 1 to 1 1/2 inches thick. Lightly season each disk with salt and pepper and place on the oiled sheet. Bake the eggplant at 450 degrees F until the slices begin turning deep brown on top, about 12-15 minutes. Remove the eggplants from the oven. Remove the slices from the baking sheet and place them on a plate to cool.
Lower oven temperature to 350 degrees F. In an 8 by 12-inch brownie pan, place the 4 largest eggplant slice evenly spaced apart. Over each slice, spread 1/4 cup of tomato sauce and sprinkle with a teaspoon of basil. Place one slice of mozzarella over each and sprinkle with 1 teaspoon grated Parmigiano. Place the smaller slices of eggplant over each of the disks and repeat with tomato sauce, basil, and the 2 cheeses. Repeat the layering again until all the ingredients are used.
Sprinkle the toasted bread crumbs over the top of the eggplant dish, and bake uncovered until the cheese is melts and the tops turn light brown, about 20 minutes. Serve immediately.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.