Bring a large pot of water to boil. Chop garlic and set aside. Cut mint leaves into thin ribbons and set aside.
In a large frying pan over medium heat, toast pine nuts, stirring, until golden, about 2 minutes. Transfer to a small bowl and set aside. In the same pan, melt 1 tbsp. butter with 2 tsp. olive oil. Increase heat to high and add half of the eggplant and 1/4 tsp. salt. Cook eggplant, stirring frequently, until soft and browned, about 5 to 10 minutes. Transfer eggplant to a plate with a slotted spoon, leaving as much of the butter and oil in the pan as possible. Repeat with remaining eggplant and 1/4 tsp. salt. Remove frying pan from heat and reserve.
Add remaining 1 tbsp. salt and fusilli to boiling water and cook until pasta is tender to the bite, 5 to 10 minutes. Drain and transfer to a large bowl.
Meanwhile, heat reserved frying pan over medium-high heat, add remaining 1 tsp. olive oil, chopped garlic, and red chile flakes, and cook until fragrant but not browned, about 1 minute. Add reserved eggplant and pine nuts. Cook, stirring, until well combined. Add eggplant mixture and reserved mint to cooked fusilli and toss to combine. Add 1/2 cup ricotta salata cheese and remaining 1 tbsp. butter. Toss until butter melts and everything is well combined. Divide among 6 plates or pasta bowls. Sprinkle with remaining 1/2 cup ricotta salata cheese and serve immediately.