A mixture of several kinds of fresh herbs is preferable, such as thyme, basil, oregano and/or tarragon.
Heat the broiler. Place a 10-inch skillet over high heat. when it is hot, add oil. Heat briefly then add zucchini, salt and pepper. Cook, stirring occasionally, until zucchini begins to soften, about 3 minutes.
Reduce the heat to medium-low; add the eggs, green onions and herbs. Cook gently, using a fork to lift the cooked portions from the sides of the pan so the uncooked centre flows underneath. Cook until the frittata is almost set, 4 to 5 minutes. Scatter the mozzarella over.
Transfer pan to the heated broiler. Broil just until the top is lightly browned and set, 30 to 45 seconds. Brush with the vinegar and sprinkle Parmesan over. Serve hot or at room temperature.