To make the yogurt spread: Place yogurt in a medium bowl and grate in garlic; mix to combine. Season with pepper, salt, and cayenne. Add dill and mix thoroughly. Set aside at room temperature.

To make Aleppo butter: Melt butter in a saucepan over medium heat; heat until bubbles begin to burst. Add chile flakes, paprika, and cumin. Stir until color is uniform. Turn off heat and let spices infuse.

To make the parsley and jalapeño oil: Grind parsley and jalapeño together in a mortar. Drizzle in olive oil and season with salt; stir to combine.

To make poached eggs: Fill a large saucepan with 2 to 3 inches of water and bring to a boil. Reduce heat to medium-low, add vinegar, and keep water at a gentle simmer. Crack an egg into a small bowl, then gently slip egg into water, holding the bowl just above surface of water. Repeat with remaining eggs. Cook until whites are firm and yolks have thickened but are not hard, 2 1/2 to 3 minutes. Remove with a slotted spoon, pat dry on a kitchen towel, and place onto a warm plate.

Dollop yogurt mixture onto serving plates. Use the back of a spoon to spread yogurt, carving ridges into the top to catch oil. Drizzle some jalapeño oil. Top with eggs and 1 to 2 spoonfuls of Aleppo butter. Sprinkle sea salt on top.