In an 8" cast iron skillet, heat olive oil over medium heat. Add in onions and saute for 3-4 minutes. Stir in shredded Brussels Sprouts, salt, and pepper. Let cook for two minutes, add in tablespoons of water, cover, and cook until Brussels Sprouts are tender.
Remove lid, make well in center, and crack eggs. Cover and let cook until egg whites have set and egg yolk is desired consistency.