Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, reserving a ladleful of pasta cooking water.
While the pasta is working, in a deep skillet, heat the EVOO, 4 turns of the pan, over medium-low heat. Add the garlic and cook for 3 minutes. Increase the heat to medium-high, add the eggplant and tomatoes and stir to coat with the oil; season with salt and pepper. Cover the pan tightly and cook, stirring halfway through, until the tomatoes burst and the eggplant is tender, about 12 minutes.
Meanwhile, in a medium bowl, mix together the ricotta and mint.
Add the pasta, reserved cooking water and the basil to the tomato-eggplant sauce and toss for 2 minutes. Serve the pasta in bowls and top with the ricotta mixture and lots of shaved parmigiano-reggiano.