Pour 4 to 6 cups sunflower oil into a large, deep skillet or Dutch oven and heat over medium-high until it reaches about 350°F (frying temperature).
Meanwhile, slice the zucchini into 1/8-inch thick rounds and spread a small handful of fresh basil leaves on a paper-towel-lined baking sheet; set aside.
Fry the zucchini in batches for about 2 minutes each, until lightly golden and beginning to crisp but not turned into chips; avoid overcrowding so the oil stays hot.
Use a spider or slotted spoon to transfer fried zucchini to the prepared baking sheet to drain and cool. When cooled to room temperature, cover and refrigerate for at least 3 hours or overnight.
Bring a large pot of salted water to a rolling boil and cook the spaghetti until al dente according to package instructions. Before draining, reserve 2 cups of the pasta cooking water; then drain the pasta.
In a large saucepan or Dutch oven over medium heat, add 2 tablespoons olive oil, 2 tablespoons butter, the smashed garlic cloves and a sprig of basil. Cook until butter is melted and garlic is fragrant, then remove and discard the garlic and basil.
Add the fried zucchini to the pan and cook over medium-low for 4–5 minutes, mashing with a wooden spoon to your desired consistency. Add a few splashes of reserved pasta water to keep the mixture moist; cook longer if you want it smoother.
Add the cooked spaghetti and 1 cup of the reserved pasta water to the zucchini mixture and toss to coat the pasta evenly.
Stir in the shredded provolone and the grated Parmesan until melted and the sauce is smooth and creamy. Add more reserved pasta water as needed to reach the desired sauce consistency.
Season with salt and pepper to taste, toss gently to combine, and serve immediately.