In a bowl, stir together the Greek-Style Yogurt with finely minced or crushed Garlic and a generous pinch of Salt; thin slightly with a splash of water if you prefer a looser consistency, then set aside at room temperature.
Bring a wide saucepan of water to a gentle simmer (small bubbles, not a rolling boil). Add a tablespoon of White Vinegar to the water to help the whites set.
Create a gentle whirlpool in the simmering water, then carefully crack and slip the Eggs in one at a time. Poach for 3–4 minutes for runny yolks or longer to your liking, then remove with a slotted spoon and drain briefly on paper towel.
While the eggs cook, melt the Unsalted Butter in a small skillet over medium heat until it foams and just begins to brown slightly — watch closely so it doesn't burn.
Remove the pan from the heat and immediately stir in the Aleppo-Style Pepper Flakes and a sprinkle of Dried Mint to bloom the flavors in the warm butter.
To assemble, spread a generous layer of the garlic yogurt on a serving plate, top with the poached eggs, and spoon the hot spiced butter over the eggs so the yolks glisten and the yogurt gets streaks of red pepper oil. Taste and adjust with more salt if needed.
Serve immediately with warm, toasted Pide for scooping — tear pieces and mop up the creamy, spicy sauce.
