Put the eggplant in a colander and toss with 1 teaspoon salt. Let stand, tossing occasionally, for 30 minutes to 1 hour. Rinse and drain well, then pat dry with paper towels.
Bring a large pot of salted water to a boil. Cook the penne according to the package directions.
While the pasta cooks, heat the oil in a large skillet over medium-high heat until it shimmers. Add the eggplant and cook, stirring occasionally, until very tender and golden brown, about 8 minutes. Add the cherry tomatoes, capers and garlic and cook, stirring often, until the tomatoes start to break down, about 2 minutes more. Transfer to a large bowl.
Add the drained pasta to the bowl with the eggplant mixture and stir in the vinegar; add salt to taste. Let stand until cooled to slightly warm or room temperature, then stir in the mozzarella and basil. Serve immediately or refrigerate for up to 4 hours.