Melt butter in a small skillet over medium-low heat and add olive oil. Stir in red pepper flakes and Aleppo pepper, letting them infuse the butter for a few minutes. Set aside once fragrant and slightly bubbly.
In a bowl, mix Greek yogurt with lemon juice, zest, garlic, salt, and pepper. For extra smoothness, I like whipping it lightly with a fork or immersion blender. This yogurt is the cool, creamy base for your eggs.
Bring water to a gentle boil and carefully add your eggs. Cook for 6–7 minutes for a soft, jammy yolk. Drain, cool briefly in an ice bath, then peel carefully.
Spread yogurt on a plate or shallow bowl, halve the eggs, and place on top. Drizzle with the warm chili butter and sprinkle chopped parsley. Serve immediately with toasted bread for dipping.