Prepare the Garlic Yogurt: Mix the yogurt with grated garlic and a pinch of salt until smooth. Set aside.
Heat the Water: Bring water to a boil in a saucepan, reduce the heat to a gentle simmer, and add the vinegar.
Poach the Eggs: Carefully slide the eggs into the water and cook for 3-4 minutes, until the whites are set.
Make the Chili Butter: Melt the butter over low heat, add the red pepper, and stir for 20-30 seconds until fragrant.
Assemble the Turkish Eggs: Spread the garlic yogurt on serving plates, add the poached eggs, and drizzle with warm chili butter.
Garnish and Serve: Add chopped dill or mint if desired. Serve immediately with fresh bread.