In a 12 to 14-inch non-stick saute pan, heat the oil over medium heat until just smoking. Add the onion and cook 7 to 9 minutes, until softened and light brown. Add the olives and tomatoes, season lightly with salt, and cook slowly for 18 to 20 minutes, until the tomatoes start to break down.
Carefully crack the eggs into the pan, keeping them whole and separated. Cover with a lid or foil and cook 3 to 4 minutes, until the whites have set but the yolk is still runny. Sprinkle basil and grated cheese over and serve from the pan on the table.