In a 10-inch enamel-lined cast-iron skillet, heat the oil over medium-heat. Add all ingredients through black pepper, and stir. Cook for 10 minutes, until vegetables are softened, stirring occasionally. Stir in the tomatoes. Bring to boiling; reduce heat. Simmer, uncovered, for 10 minutes or until slightly thickened, stirring occasionally.
Break the eggs into a small dish (being careful not to break the yolks) and slip the eggs into the tomato mixture. Reduce the heat. Cook, covered, 5 to 10 minutes or until the whites are completely set and the yolks are of desired doneness. Sprinkle cheese and oregano on top.