Set the Greek yogurt out while you prep. Stir it with the grated garlic, lemon juice, and salt until smooth. Keep it off the heat; boiling the yogurt will make it split.

Bring a large pot of salted water to a boil. Cook the pasta until just al dente. Reserve 1/2 cup pasta water, then drain. Let the pasta steam for about 60 seconds before assembling so it is hot but not bubbling.

Fill a deep saucepan with water, add the vinegar, and bring it to a gentle simmer. Crack each egg into a ramekin, slide it into the water, and poach for 3 to 4 minutes, until the whites are set and the yolks are still soft. Lift out with a slotted spoon.

Melt the butter in a small light-colored skillet over low heat. Stir in the Aleppo pepper, paprika, and salt for about 30 seconds, just until the butter turns red-orange and smells fragrant. Do not let the spices burn.

Spoon the garlic yogurt into four shallow bowls. Divide the warm pasta over the yogurt, adding a splash of reserved pasta water if needed. Top each bowl with a poached egg, spoon over the chili butter, and finish with dill. Serve right away.