In a food processor, combine feta cheese, Greek yogurt, cream cheese, olive oil, lemon juice, and black pepper. Blend until smooth, about 2–3 minutes.

In a medium saucepan, bring about 3 inches of water to a gentle simmer. Add a splash of vinegar. Create a whirlpool and slip in each egg. Cook for 3–4 minutes.

In a small skillet, melt butter over medium-low heat, then add garlic, Aleppo pepper, and smoked paprika. Stir for about 1 minute.

On a serving platter, spread whipped feta, place poached eggs on top, and drizzle with chile butter.

Serve immediately while warm, paired with crusty bread.