Using a vegetable peeler, peel half of the skin off the eggplant in long strips, leaving a strip of skin between peeled sections. (The eggplant will look striped.) Chop the eggplant and toss with salt. Let the eggplant drain on a kitchen towel while you start the sauce.
Bring a large pot of water to a boil for the pasta.
Heat a large saucepot or skillet over medium-high to high. Add the olive oil, two turns of pan. Add the sausage and cook, stirring occasionally and breaking the meat up with a wooden spoon, until browned, about 5 minutes. Squeeze the excess liquid from the eggplant and add to the saucepot. Add the onion and garlic. Cover partially and cook, stirring occasionally, until the vegetables soften, about 5 minutes. Add the tomatoes and cook, stirring occasionally and breaking the tomatoes up with the spoon, until the sauce thickens, about 5 minutes. Wilt in the basil and stir in the cherry peppers and brine.
Salt the boiling water and add the pasta. Cook until 1 minute shy of the package directions. Reserve 1 cup of the cooking water, then drain.
Add the pasta, cooking water, a fat handful of cheese, and a swirl of EVOO and toss until the pasta is coated. Serve in shallow bowls and pass more cheese at the table.