Slice the zucchini into thin rounds.
In a wide pan, heat about ½ inch of olive oil over medium heat. Fry the zucchini in batches until lightly golden. Transfer to a paper towel-lined tray, season immediately with salt, and toss with the torn basil. Let the zucchini rest for at least 30 minutes so the flavors can develop.
Bring a large pot of generously salted water to a boil and cook the spaghetti until just shy of al dente. Reserve 1½ cups of the pasta water before draining.
In a large sauté pan, warm a few tablespoons of olive oil and gently infuse it with the smashed garlic clove over low heat. Once fragrant and lightly golden, remove and discard the garlic.
Add the rested zucchini along with a splash of the reserved pasta water. Let it simmer gently for 2–3 minutes, stirring so some of the zucchini begins to break down into the oil.
Add the drained spaghetti and toss continuously. Off the heat, gradually add the Parmigiano, and additional pasta water a little at a time, tossing vigorously until a smooth, glossy sauce forms and coats every strand of pasta.
Serve immediately topped with additional grated cheese and fresh basil.