In a bowl, add the yogurt, dill, mint, garlic, salt and pepper. Mix and set aside.
Fill a pot with water and bring it to a boil. In two small ramekins or bowls, crack open the eggs, two in each one of them.
Add a spoonful of vinegar and create a swirl with the help of a spoon, right before adding the eggs. Slowly pour the eggs from the bowls, right in the middle of the vortex. Let them cook for 3 minutes, so that the yolk remains soft but the whites are firm.
In a small saucepan, on medium heat, add the butter. When the butter is melted, add the chili flakes, paprika, salt and pepper. Simmer for 1 minute, then remove it from the heat and mix well.
To assemble, divide the Greek yogurt in two and transfer it to the plates. Carefully place two poached eggs on each plate. Pour the butter sauce and sprinkle with pomegranate seeds and fresh mint leaves.
Serve with toasted baguette, bread slices or pita.