Bring a large pot of water to boil for the pasta.
Heat a grill pan, griddle or grill over medium-high.
Slice the eggplant no more than 1/2 inch thick. Arrange the eggplant slices on a clean kitchen towel. Sprinkle all over with salt and let sit a few minutes to drain out some of the juices. Pat the eggplant dry. In a medium bowl, whisk 3 tbsp. EVOO, the vinegar, honey, crushed garlic and 1/3 cup water. Add the eggplant; turn to coat.
Grill the eggplant until tender, 3 to 5 minutes on each side, brushing with more of the EVOO mixture if dry. Let cool and chop into bite-size pieces.
In a large bowl, combine the tomatoes, basil, parsley, chile, the remaining 3 tbsp. EVOO and the pasted garlic. Add any leftover EVOO mixture from the eggplant, discarding the crushed garlic; toss to coat.
Salt the boiling water. Add the pasta; cook to al dente. Drain the pasta, then run it under cold water. Drain well, then add to the tomato mixture. Add the eggplant and cheese; toss. Season with salt and pepper.