Heat olive oil in a frying pan over medium heat until it shimmers and you can smell it.
Add the sliced peppers to the pan and cook for about 3-4 minutes, stirring occasionally, until they soften and begin to caramelize slightly, filling the air with a sweet aroma.
Toss in the diced tomatoes and sprinkle with cumin, then stir to combine. Cook for another 5-6 minutes, allowing the tomatoes to soften and create a fragrant, slightly thickened sauce.
Make small wells in the sauce and crack the eggs directly into them. Gently nestle the eggs into the sauce without breaking the yolks.
Reduce the heat to low and cover the pan briefly, cooking the eggs for about 3-4 minutes until they are just set but still creamy.
Remove the pan from heat, season with salt and freshly ground pepper to taste, and let everything rest for a minute so the flavors meld.
Serve the menemen hot straight from the pan, with crusty bread on the side for dipping and scooping up the saucy, soft eggs and vegetables.