In a large cast-iron skillet, heat 2 tablespoons olive oil over medium-high heat. Add the onions and rosemary to the skillet, season with salt and pepper, and cook for 5 minutes or until they are soft, stirring. Remove from the heat and set aside.
Add the eggs to a medium bowl. Season with salt and black pepper and whisk until combined. Set aside.
In a medium cast-iron skillet, heat the additional 2 tablespoons olive oil over medium-high heat. Pour the eggs into the skillet and cook for 4-5 minutes or until just set. Spoon the onions evenly on one side of the omelet and fold the other side over.
Transfer the omelet to a serving plate. Sprinkle the oregano and black pepper over the top. Serve immediately.