1. Preheat oven to 500º F. 
2. Place the eggplant cubes into a medium bowl and drizzle with 4 tablespoons of olive oil.  Add 1 teaspoon of kosher salt and 1 teaspoon black pepper and toss to combine. Transfer eggplant to 2 baking sheets and bake, turning twice with a spatula, until soft and caramelized, about 20 minutes. Set aside. 
3. Heat the remaining 3 tablespoons olive oil in a large pot or pan over medium heat. Add the diced onions and cook, stirring, until soft and translucent, but not browned, about 6-8 minutes. 
Add the crushed red pepper and garlic and cook, stirring, until garlic softens and becomes fragrant, about 1 minute. 
4. Add the crushed tomatoes, half the basil, 1 teaspoon salt and bring to a boil, then lower heat to a slow simmer. Meanwhile, add 2 tablespoons of kosher salt to a large pot of water and bring to a boil.
5. Add the rigatoni and cook, stirring occasionally, until al dente, about one minute less than the package directions. Drain pasta, reserving 1½ cups cooking liquid, and transfer pasta to the simmering tomato sauce along with the capers. 
6. Add 1 cup pasta cooking liquid and stir for 1 minute, incorporating cooking liquid with the sauce.
Turn off heat and allow pasta to rest for 5-8 minutes absorbing sauce, stirring occasionally. Gently stir in the reserved eggplant and toss to combine. 
7. Add the remaining half cup pasta cooking liquid if pasta needs more sauce. Stir in remaining basil and check for seasoning, adding more salt if necessary. 

8. To serve, transfer pasta to a platter and top with grated cheese, with an additional bowl of grated cheese to pass around. This pasta is actually better served warm or at room temperature rather than piping hot.