Squeeze grated zucchini in paper towels to release as much water as possible.
Spread zucchini on fresh paper towels and sprinkle with salt; let sit for 30 minutes to release additional water; squeeze again.
Mix Parmesan cheese, eggs, green onions, garlic, basil, nutmeg, onion powder, and black pepper in a bowl; add zucchini. Stir to combine.
Heat butter and olive oil in a frying pan over medium heat. Form golf ball-sized zucchini patties and place in hot frying pan. Fry until browned, 2 to 3 minutes per side.