Heat olive oil in a skillet over medium heat and sauté the diced onion until golden.
Add diced bell pepper to the onion, stir, and continue cooking until the pepper pieces become soft.
Crack the eggs into a deep bowl and whisk them lightly with a fork – just enough to blend the yolks and whites together.
Add the diced tomatoes (along with their juices) to the sautéed onion and pepper. Stir the vegetables, cover the skillet with a lid, and simmer over low heat until the liquid reduces by half, about 8–10 minutes. If the tomatoes aren't very ripe or flavorful, you can add ½–1 tablespoon of tomato paste. Season with salt and black pepper to taste.
Pour the beaten eggs into the skillet. Over low heat, stirring gently every 5–10 seconds, cook the mixture for another 1–2 minutes.
Turn off the heat, add the finely crumbled or grated cheese, and stir. Immediately cover the skillet with a lid and let the menemen rest for 1–2 minutes.
It's important that the eggs remain moist and slightly runny when you turn off the heat. Sometimes, the eggs aren't even cooked with the vegetables – they're simply added to the skillet and the heat is turned off immediately. Since the sauce is hot, the eggs will finish cooking under the lid using the residual heat of the cast-iron skillet.
You can garnish the menemen with fresh basil or parsley leaves. Enjoy!