In a small bowl, stir together the yogurt, grated garlic, ¼ teaspoon salt, and the chopped dill until smooth. Divide the mixture between two shallow serving bowls, spreading it out to create a thick bed for the eggs.

Meanwhile, melt the butter in a small skillet over medium heat until it foams, 1 to 2 minutes. Stir in the Aleppo pepper and paprika, and cook 20–30 seconds, just until fragrant. Remove from heat and set aside.

Crack each egg into a small fine mesh sieve over a small bowl. Allow the liquidy egg whites to drain. Then transfer each egg to four individual ramekins.

Fill a deep skillet with 3 inches of water and bring it just to a gentle simmer (small bubbles, not a rolling boil). Stir in the vinegar. Swirl the water to create a vortex, gently slide in each egg, and cook 3 minutes for a runny yolk (3½ to 4 minutes for medium). Lift out with a slotted spoon, letting excess water drain.

Nestle two poached eggs onto the yogurt in each bowl. Drizzle the hot chili-butter evenly over the eggs and yogurt and garnish with extra dill. Serve warm with crusty bread or pita for scooping.