Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, then rinse under cold water.
While the pasta is working, heat a grill or grill pan to medium-high. Brush the eggplant with EVOO, season with salt and pepper and grill until tender, 6-8 minutes. Let cool for 5 minutes, then cut into small cubes and place in a large bowl.
Add the tomatoes, basil and garlic to the eggplant. Pour in the 1/4 cup EVOO and season with salt and pepper. Add the pasta and cheese and toss to combine.