In a large mixing bowl, combine Greek yogurt, olive oil, lemon juice, grated garlic, chopped dill, chopped mint, and kosher salt. Whisk until smooth and creamy. Set aside to let flavors meld.

Fill a medium saucepan with at least 3 inches of water and add distilled white vinegar. Place over medium heat and bring water to just below a gentle simmer. Crack each egg into a small ramekin or bowl and carefully lower them one at a time into the simmering water. Poach for about 3 minutes until egg whites are set but yolks remain runny.

Using a slotted spoon, lift the poached eggs from the water and transfer them to a plate lined with paper towels or a kitchen towel to absorb excess moisture. Repeat with remaining eggs.

In a small pan over medium heat, melt unsalted butter. Stir in Aleppo pepper and cook for about 1 minute until the butter is fragrant.

Spoon the herbed yogurt into shallow serving bowls. Place two poached eggs on top of each portion of yogurt. Drizzle the warm Aleppo butter sauce over the eggs. Garnish with additional fresh dill or mint, and sprinkle flaky sea salt and freshly ground black pepper over the eggs. Serve immediately with pita or crusty bread for dipping.