Place sunflower oil in a deep fryer and heat to 350 degrees F (175 degrees C). Slice zucchini into 1/8-inch rounds.

Add some zucchini to the hot oil and deep-fry until lightly browned, 4 to 5 minutes. Transfer to a single layer on paper towels to drain and repeat to cook remaining zucchini. Let zucchini cool to room temperature.

Transfer cooled zucchini to a bowl. Cover and refrigerate, 8 hours to overnight.

Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until it’s 2 minutes away from being tender yet firm to the bite, 9 to 10 minutes.

As you are cooking the spaghetti, heat olive oil in a skillet over medium heat. Add zucchini and salt and cook, stirring occasionally, until it starts to release juices, soften, and break down, 3 to 5 minutes. Add a couple splashes of the pasta water to the zucchini, but don’t overdo it. Break zucchini into small pieces with the edge of a spoon or spatula. Reduce heat to low and toss in basil and butter; stir until butter melts.

Grab spaghetti with tongs and transfer it right into the sauce; stir until coated. Mix in Pecorino Romano and Parmigiano-Reggiano cheeses. Add a final splash of pasta water and adjust salt if needed.

Serve with more grated cheese and fresh basil.