1. Shred zucchini and onion. Pour olive oil into a 10- to 12-inch frying pan over medium-high heat. Add vegetables, salt, and pepper and cook, stirring frequently, until vegetables are soft and any liquid has evaporated, about 8 minutes.
2. Transfer mixture to a large bowl and stir in bread crumbs, egg, cheese, parsley, and mint until incorporated. Let stand until cool enough to handle, about 15 minutes. With well-floured hands, form mixture into patties about 2 inches wide and 1/2 inch thick.
3. Wipe pan clean and heat vegetable oil over medium heat. When oil is hot, add patties, three or four at a time. Cook, turning once, until golden brown on both sides, about 6 minutes total. Reduce heat as needed to maintain a constant temperature. With a spatula, transfer to a paper towel-lined baking pan; keep warm in 200° oven. Serve hot.