In a medium mixing bowl stir together the Greek yogurt, garlic, lemon zest and white pepper until thoroughly combined. Cover and refrigerate until ready to use.
For the chili oil:
Heat a medium sauté pan over medium heat, add the butter, oil and Aleppo pepper. Heat until the butter is melted and the oil turns an orange-red color, about 5 minutes. Remove from heat and let cool completely.
For the Za'atar-spiced pitas:
Heat oven to 350ºF. Arrange pitas on a baking sheet. Brush each pita with butter and sprinkle with Za'atar.Bake until golden brown and crisp, about 8 minutes.
For serving:
Divide the Greek yogurt among plates. Top each with a poached egg and drizzle each with some of the chili oil. Garnish with flake salt followed by mint and parsley. Serve with the Za'atar spiced pita bread.
Notes
For the creamiest base, let the Greek yogurt sit at room temperature for 10–15 minutes before assembling so it’s silky and spreadable—this helps it blend beautifully with the warm chili oil.