Bring a large pot of water to boil. Meanwhile, trim and discard ends of zucchini. Cut each zucchini into 3- to 4-in. lengths; cut each length into 1/4- to 1/2-in.-thick matchsticks and set aside. Chop garlic and set aside. Cut basil 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 zucchini and 1/4 tsp. salt. Cook zucchini, stirring frequently, until soft and browned, about 5 minutes. Transfer zucchini to a plate with a slotted spoon, leaving as much of the butter and oil in the pan as possible. Repeat with remaining zucchini 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 and the chopped garlic, and cook until fragrant but not browned, about 1 minute. Add reserved zucchini and pine nuts. Cook, stirring, until well combined. Add zucchini mixture and reserved basil to cooked fusilli and toss to combine. Add 1/2 cup parmesan 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 parmesan and serve immediately.
Note: Nutritional analysis is per serving.