Preheat the oven to 375°F.
In a large ovenproof skillet, heat the olive oil until shimmering. Add the onion, season with salt and pepper and cook over moderate heat, stirring occasionally, until softened, about 5 minutes. Add the tomatoes and cook, stirring occasionally, until some of them burst, 5 to 7 minutes. Add the bell peppers and cook, stirring occasionally, until the peppers are softened and all of the tomatoes have burst, about 10 minutes. Remove from the heat and stir in ¼ cup of the mint and 1 ½ teaspoons of the za'atar. Season the sauce with salt and pepper.
Using a spoon, make 4 wells in the tomato sauce, then crack an egg into each one and season with salt and pepper. Transfer the skillet to the oven and bake until the egg whites are just set and the yolks are still runny, 10 to 12 minutes.
Meanwhile, in a small bowl, mix the yogurt with 1 tablespoon lemon juice, 1 teaspoon lemon zest and the remaining 2 teaspoons mint and ½ teaspoon za'atar. Season with salt and pepper and garnish with a pinch more za'atar. Drizzle the eggs with olive oil and garnish with a little more mint. Serve with the yogurt.
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