Begin by grating the zucchini using a box grater, ensuring you create fine shreds for optimal texture. After grating, place the zucchini in a cheesecloth or kitchen towel and squeeze out as much moisture as possible.

In a mixing bowl, combine the dry zucchini, one beaten egg, freshly grated Parmesan cheese, cornstarch, salt, and pepper. Stir the mixture until it reaches a dough-like consistency.

Heat a generous tablespoon of garlic butter in a nonstick skillet over medium heat. Once the butter is melted, scoop the zucchini mixture into the skillet, forming two evenly-sized patties. Cook each patty for 3–4 minutes until golden brown, then flip carefully.

Sprinkle shredded cheddar cheese on top of one patty, then place the other patty on top to form a sandwich. Lower the heat and cook for an additional 3 minutes, pressing gently to help the cheese melt.

After the cheese has melted and the exterior is crispy, carefully remove the sandwich from the skillet. Let it sit for about 60 seconds before slicing. Serve hot.