In a medium skillet, heat 2 tablespoons of the oil. Add the onion and ½ teaspoon of salt and cook until translucent, 4 to 5 minutes, stirring occasionally.
In a large bowl, massage the kale with ½ teaspoon of salt until tender.
Preheat the oven to 375°F., and warm a cast iron or other oven-safe skillet for 5 minutes.
In a separate bowl, whisk the eggs and fold in the kale, onion, cheese, pepper and remaining sea salt.
Coat the warmed skillet with the remaining oil. Pour in the egg mixture. Place the skillet in the oven and cook until set, about 15 minutes.