Heat the olive oil in a wide skillet and add the garlic clove until fragrant. Remove the garlic, then gently fry the zucchini slices until soft and lightly golden. They should be tender, not crispy.
Bring a large pot of salted water to a boil and cook the spaghetti until just al dente. Before draining, reserve some pasta water to help adjust the sauce later if needed.
Transfer about one-third of the cooked zucchini into a bowl and mash or blend it with a little pasta water. This forms the creamy foundation of the sauce without using cream.
Add the hot spaghetti to the zucchini purée along with the remaining zucchini, grated cheese, and cold butter. Toss well until the pasta is glossy and evenly coated.
Season with salt and black pepper, then add torn basil leaves. Serve immediately while the pasta is warm and creamy.