Warm the olive oil and butter in a large nonstick skillet over medium heat. Add the onion, if using, and cook for 2 to 3 minutes until softened but not browned.
Add the diced pepper and cook for 3 to 4 minutes, stirring occasionally, until it begins to soften.
Stir in the tomatoes, salt, black pepper, and optional Aleppo pepper. Cook for 4 to 5 minutes, stirring now and then, until the tomatoes break down into a saucy mixture and most excess liquid has reduced.
Crack the eggs directly into the skillet. Use a spatula to gently break the yolks and fold the eggs through the tomato mixture. Cook over medium-low heat, stirring slowly, until the eggs are softly set and cooked to your preferred doneness, about 2 to 3 minutes.
Remove the skillet from the heat, sprinkle with parsley if desired, and serve right away with warm crusty bread, pita, or toast.