Perciatelli is a long, tubular pasta; spaghetti may be substituted.
Preheat oven to 400 degrees. In a medium bowl, combine eggplant, 1 tablespoon oil, 1/4 teaspoon salt, and 1/8 teaspoon pepper; toss to combine. Transfer to a baking pan in a single layer. Place all tomatoes and garlic cloves in a second pan. Transfer both pans to oven. Roast eggplant until brown and tender, about 40 minutes; roast garlic and tomatoes until garlic is soft and tomatoes are shriveled, about 30 minutes. When cool enough to handle, squeeze soft garlic from cloves.
Meanwhile, place peppers over a gas burner with a high flame; roast until charred on all sides, turning as needed, about 5 minutes. Transfer to a metal bowl; cover with plastic wrap for 20 minutes. Using a paper towel, rub charred skin off peppers. Remove and discard seeds and stems. Place 1 red and 1 yellow pepper, and any liquid collected in bowl, in the bowl of a food processor. Add roasted garlic, plum tomatoes, vinegar, cayenne, remaining tablespoon olive oil, and remaining 1/2 teaspoon salt and 3/8 teaspoon pepper; process until smooth. Cut remaining peppers into 1/4-inch-thick strips; set aside
Fill a large stockpot with salted water; bring to a boil. Place pasta in water; boil until pasta is al dente, about 11 minutes. Drain in a colander; transfer to a serving bowl. Pour puree over pasta; toss to combine. Add basil, roasted eggplant, cherry tomatoes, and reserved pepper strips; toss. Serve garnished with grated Parmesan cheese, if desired.