Preheat broiler. Break eggs into a medium bowl. Whisk together eggs, the water, green onions, chopped cilantro, salt, and pepper; set aside.
In a 12-inch broiler-proof skillet with flared sides, heat olive oil over medium heat. Add egg mixture. Lift edges of egg mixture as it sets, allowing uncooked mixture to flow underneath. When edges begin to set (top is soft but not runny), place eggs under broiler, 3 to 4 inches from heat. Broil for 1 minute or just until set. Top with cheese; return to broiler just until cheese is melted.
Cut frittata into wedges. To serve, top with carrot shreds, edamame, additional cilantro, and, if desired, shaved Parmesan cheese.