To extract the moisture from the zucchini, spread the zucchini out on a tray and sprinkle with 1 teaspoon salt. Set aside. While the zucchini sits, make the balsamic caramelized onions: In a medium skillet over medium heat, melt 1 tablespoon of butter. Add onion slices and stir to coat with butter. Sprinkle the onions with reserved ½ teaspoon salt. Stir until onions cook down and begin to turn golden, about 5 minutes. Add 2 teaspoons sugar and continue to stir until onions darken to a medium golden color and turn translucent, about 5 more minutes. Add ¼ cup balsamic vinegar, scrape up the bits from the bottom of the pan and turn heat to low. Heat, stirring occasionally, until most of the liquid absorbs, about 5-7 more minutes. Remove from heat and set aside.
Preheat oven to 375 degrees. In small batches, place the zucchini shreds in a clean cloth and squeeze the water out. Repeat as needed. In a large skillet over medium heat, melt reserved 2 tablespoons of butter. Add the zucchini shreds and cook until they begin to turn slightly golden. Add corn and stir until heated through. Transfer zucchini and corn to a large bowl and let cool. If the mixture is too hot for these next steps, the ingredients clump together and will result in uneven cooking. Add the caramelized onions, shredded mozzarella, 2 tablespoons cilantro and 6 eggs. Gently stir until uniform. Transfer the mixture to a greased 9.5 inch glass pie dish. Cover loosely with greased foil, grease side down. Bake for 25-30 minutes until there is no liquid when a knife is inserted into the center of the egg bake. Remove foil and broil until the cheese on top is brown and bubbly, about 3-5 minutes. Remove from oven and cool. Garnish with cilantro just before serving.
Serve warm with fruit for breakfast, or with a side salad for lunch or dinner.