In a large lidded skillet, heat 2 tbsp of the oil over medium heat. Stir in tomatoes and eggplant. Cook, covered, 15 minutes. (If skillet does not have a lid, try covering tightly with aluminum foil or a baking sheet.) Stir in garlic and 1/4 tsp of the salt; cover and cook another 10 to 15 minutes until tomatoes are slightly wilted.
Meanwhile, bring a pot of lightly salted water to a boil. Cook fusilli according to package directions, reserving 3/4 cup of the pasta water. Using a slotted spoon, transfer fusilli to skillet with tomatoeggplant mixture. Stir in 1/2 cup of the pasta water, 3/4 cup of the basil, 3/4 cup of the ricotta salata, the remaining 2 tbsp oil and remaining 1/2 tsp salt. (If pasta seems dry, stir in 1/4 cup pasta water.) Garnish with remaining 1/4 cup basil and 1/4 cup ricotta salata.