1. Preheat broiler with a rack set 3 to 4 in. from heat. In an ovenproof frying pan, cook onion and mushrooms in oil over medium-high heat, stirring often, until browned, 8 to 12 minutes. Add kale and 1/4 cup water; cook, stirring occasionally, until wilted and tender, 4 to 6 minutes.
2. Meanwhile, whisk eggs. Add 1/2 cup cheese, the salt, pepper, and olives.
3. Reduce kale pan's heat to medium; level veggies. Pour in eggs, spreading out olives. Cook until frittata sides look set (top will be wet), 2 to 3 minutes. Sprinkle on 1/2 cup parmesan. Broil until lightly browned and firm, 3 minutes. Sprinkle with chile flakes.