In a medium bowl, combine Greek yogurt with olive oil, lemon juice, minced garlic, chopped dill, and kosher salt. Whisk until smooth and creamy, then set aside.

Fill a medium saucepan with about 3 inches of water and add distilled white vinegar. Heat until it simmers gently. Crack eggs into small bowls and slide them into the simmering water, poaching for about 3 minutes.

Melt unsalted butter in a small saucepan over low heat. Stir in Aleppo pepper and let it infuse for about 1 minute.

Spread the herbed yogurt in shallow bowls, place the poached eggs on top, drizzle with warm spiced butter, and garnish with fresh herbs if desired.