In a medium bowl, mix the yogurt with the garlic and ½ teaspoon of the salt. Stir until smooth and spreadable. Let it sit at room temperature while you prepare the rest, so it softens and will cradle the eggs instead of stiffening around them.

In a small saucepan, melt the butter over medium heat until it begins to foam. Add the Aleppo pepper and swirl gently. The butter should turn a deep red and smell fragrant, but the spices should not darken too much. Remove from the heat and set aside to infuse.

Fill a wide pot with about 3 inches of water. Bring it to a gentle simmer over medium heat, then add the vinegar and remaining ½ teaspoon salt. You want small bubbles forming on the bottom, not a rolling boil, so the egg whites stay close to the yolks.

Crack one egg into a small bowl. Stir the water in one direction to create a soft whirlpool, then slide the egg into the center. Let it cook for about 3 minutes, until the whites are set and the yolk feels soft when lifted with a slotted spoon. Transfer to a paper towel to drain. Repeat with the remaining eggs, one at a time.

Spread the garlicky yogurt onto a large plate or shallow bowl in an even layer. Nestle the poached eggs on top. Spoon the warm chili butter over the eggs and yogurt, letting the colored butter run into small pools. Sprinkle with chopped dill.

Serve immediately with pita or warm crusty bread for scooping, while the eggs are still warm and the butter has not firmed.