Place the shredded zucchini in a colander and sprinkle with about 1/4 teaspoon salt. Let the salted zucchini sit for about 30 minutes. Then rinse well with cold water and press out as much of the water as possible. Blot dry with a lint-free towel or paper towels.
Preheat the waffle iron to medium and the oven to its lowest setting.
In a large bowl, whisk together the egg, milk, and 1/4 cup of the grated Parmesan. In a small bowl, combine the flour with about 1/4 teaspoon salt and 1/4 teaspoon freshly ground black pepper. Mix the contents of the small bowl into the large bowl and then add the zucchini and toss until everything is well-combined.
Coat the waffle iron with nonstick spray or brush it with a neutral oil. Place rounded tablespoons of the batter on the waffle iron, leaving room for the fritters to spread slightly. Close the lid and cook until lightly browned, about 3 minutes. (You may need to turn your waffle iron up or down slightly, depending on how powerful it is.) Transfer finished fritters to the oven to keep warm while you cook the remaining fritters. Serve topped with the remaining 1/4 cup Parmesan.