Make the Aleppo butter. Melt the butter over medium heat in a small saucepan. Add the olive oil and Aleppo pepper. Stir until evenly combined and a bit foamy before removing from heat. Set aside.
Poach the eggs. Fill a large sauté pan 3/4 of the way with water. Add the vinegar and bring everything to a simmer. Swirl the water in a circle using a spoon then gently add the eggs to the center of the vortex. Once the whites are solid transfer the eggs to a paper towel lined plate to dry.
Serve the eggs over yogurt and top with Aleppo butter. Add a half cup of yogurt to each plate before topping with an egg and a generous drizzle of the Aleppo butter sauce.