Heat sunflower oil in a deep fryer to 350°F (175°C) and slice the zucchini into thin 1/8-inch rounds.
Fry the zucchini in batches until they turn golden brown, approximately 4 to 5 minutes. Once cooked, place them on paper towels to drain excess oil and allow to cool.
Once cooled, store the zucchini in a bowl, cover it, and let it chill in the refrigerator for at least 8 hours or overnight.
In a large pot, bring lightly salted water to a boil and cook spaghetti until it is just underdone, about 9 to 10 minutes, stirring occasionally.
While the spaghetti cooks, heat olive oil in a skillet over medium heat. Add the refrigerated zucchini and a pinch of salt, cooking until it begins to soften and release its juices, about 3 to 5 minutes. Use a splash of pasta water to help break down the zucchini into smaller pieces.
Reduce the heat to low, then add fresh basil and butter, stirring until the butter is fully melted and incorporated.
Using tongs, transfer the spaghetti directly into the skillet with the zucchini sauce. Toss to ensure the pasta is well-coated, then stir in grated Pecorino Romano and Parmigiano-Reggiano cheeses. If needed, add a bit more pasta water and adjust the seasoning for salt.
Serve immediately, garnished with additional grated cheese and fresh basil.