Place sunflower oil in a deep fryer and heat to 350°F (175°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 paper towels to drain and repeat with remaining zucchini. Let zucchini cool to room temperature.

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

Bring a large pot of lightly salted water to a boil. Cook spaghetti until 2 minutes away from being tender yet firm to the bite, about 9 to 10 minutes.

As you cook the spaghetti, heat olive oil in a skillet over medium heat. Add zucchini and salt, cooking until it starts to release juices, soften, and break down, about 3 to 5 minutes. Add pasta water sparingly and break the zucchini into small pieces.

Reduce skillet heat to low and toss in basil and butter until melted.

Transfer cooked spaghetti into the zucchini sauce. Mix in Pecorino Romano and Parmigiano-Reggiano cheeses, adding more pasta water if needed. Adjust salt to taste.