How to Make It
Fill a large skillet with water to a depth of 2 1/2 inches. Bring to a simmer over medium-high.
Fill a medium saucepan with water to a depth of 2 inches; bring to a boil over high. Place yogurt in a medium heatproof bowl, and place bowl on top of saucepan. (Make sure base of bowl doesn’t touch the water.) Stir in garlic and salt. Cook, stirring, until yogurt mixture reaches about 100°F and is the consistency of lightly whipped cream. Remove pan from heat, leaving bowl over saucepan.
Melt butter in a small saucepan over medium, stirring occasionally, until just beginning to turn hazelnut brown, about 4 minutes. Remove from heat, and stir in oil and Aleppo pepper. Set aside.
Crack 1 egg into a fine wire-mesh strainer over a small bowl. Gently swirl strainer until watery part of egg white drips into bowl, 20 to 30 seconds. Gently tip egg from strainer into a small cup or ramekin. Add 1 teaspoon lemon juice. Repeat procedure with remaining egg and remaining 1 teaspoon lemon juice.
Gently slide eggs, 1 on each side of skillet, into the simmering water in the large skillet. Reduce heat so there is no movement in the water, and poach eggs until whites are set and yolks are still runny, 3 to 4 minutes. Using a slotted spoon, remove the eggs to a paper towel–lined plate.
Divide the warm creamy yogurt mixture between 2 shallow bowls. Top each with a poached egg, and pour the peppery butter around and slightly over the yogurt. Sprinkle with chopped dill, and serve with toasted bread.