With a mortar and pestle, smash the peeled garlic cloves with a pinch of salt until smooth. Mix the garlic into the yogurt and transfer to individual serving plates.
Crack each egg into separate, small bowls.
In a medium pot, combine the water, vinegar, and pinch of salt. Bring to a boil. Once the water is boiling, reduce heat to medium and stir the water in circles to create a vortex. Gently add each egg, one at a time, into the center of the vortex. Cook them in batches if needed. Poach for about 3-4 minutes before removing each egg with a slotted spoon. Place one to two eggs over each plate of garlicky yogurt.
While the eggs are cooking, melt the butter over medium heat in a small pan. Once it begins to sizzle, stir in the pepper and paprika and remove from heat.
Drizzle the spiced butter over the eggs and yogurt. Sprinkle with fresh mint and/or dill and a pinch of salt. Serve immediately with slices of crusty bread or Pide.