Gently squeeze excess moisture from the zucchini using a clean kitchen towel.
Next, in a medium bowl, mix the zucchini with an egg, Parmesan, green onions, and cornstarch. Then season the mixture with some salt and pepper according to your taste.
Now take a large skillet and pour enough vegetable oil to layer the bottom of the pan. Scoop about 1/4 of the zucchini mixture onto one side of the pan and shape it into a small square.
You can place two zucchini patties of the same size in the same skillet and cook them over medium heat and cook them together.
Cook until the patties are lightly golden on both sides, about 4 minutes per side.
Remove the patties from the heat to drain on paper towels, and repeat with the remaining zucchini mixture. Wipe the skillet clean before shallow frying the next batch of patties.
Preparing the second batch, top both the patties with shredded cheese, then place the previously cooked patties on top to form two sandwiches. Cook until the cheese has melted, about 2 minutes per side.
Cut in half and serve immediately.