Preheat the oven to 350 degrees F.
Wash and towel dry the eggplant. Slice the eggplant horizontally about 1/4-inch thick. Place the slices in a large colander, sprinkle with salt and set aside to rest about 30 minutes. Drain and rinse the eggplant and dry on towels.
In a saute pan, heat the extra-virgin olive oil until just smoking. Press the drained eggplant pieces into the seasoned bread crumb mixture and saute until light golden brown on both sides. Repeat with all of the pieces. On a cookie sheet lay out the 4 largest pieces of eggplant. Place 2 tablespoons of tomato sauce over each piece and place a thin slice of mozzarella on top of each. Sprinkle with Parmigiano and top each with the next smallest piece of eggplant, then sauce then mozzarella. Repeat the layering process until all the ingredients have been used, finishing again with the Parmigiano. Place the pan in the oven and bake until the top of each little stack is golden brown and bubbly, about 15 minutes.
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.
Recipe courtesy of Mario Batali