Prepare mezzelune according to package directions. While mezzelune cooks, place a small skillet over medium heat and pour just enough sunflower oil in it to coat the bottom. Set a paper-towel-lined plate close to the skillet to drain and dry the fried basil. When the oil is hot, use tongs to carefully lower basil leaves into the skillet. Fry for 5–15 seconds, then flip them and fry another 5–15 seconds. Remove fried basil from skillet and set on paper towels to dry.
Plate the cooked mezzelune and generously drizzle with olive oil. Garnish with grated parmesan and fried basil, and enjoy immediately.