Pour 4–6 cups sunflower oil into a large, deep frying pan or pot and heat over medium-high until it reaches about 350°F (use a thermometer) or is shimmering.
Slice the zucchini crosswise into 1/8-inch-thick medallions and spread a small handful of basil leaves on a paper-towel-lined baking sheet; set the sheet nearby.
Fry the zucchini in batches so the oil temperature stays hot, about 1½–2 minutes per batch, until lightly golden and beginning to crisp but not browned into chips.
Use a spider or slotted spoon to transfer fried zucchini to the prepared baking sheet to drain and cool. Repeat until all zucchini is fried.
Once cooled to room temperature, cover the baking sheet and refrigerate the zucchini 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 directions. Before draining, reserve 2 cups of the pasta cooking water, then drain the pasta.
Warm a large saucepan or Dutch oven over medium and add 2 tablespoons olive oil and 2 tablespoons butter with the smashed garlic and a sprig of basil; when butter is melted and fragrant, remove and discard the garlic and basil.
Add the fried, cooled zucchini to the pan and cook over medium–low for 4–5 minutes, mashing with a wooden spoon to your desired texture; add a few splashes of reserved pasta water to keep it moist and continue cooking briefly if you want it very soft.
Add the drained spaghetti and 1 cup of the reserved pasta water to the pan and toss to combine and coat the pasta with the zucchini mixture.
Stir in the shredded provolone and grated Parmesan, mixing until the cheeses melt into a smooth, creamy sauce; add more reserved pasta water as needed to reach desired creaminess.
Season with salt and pepper to taste and serve immediately.
