Make the yogurt sauce: Mix together the yogurt, garlic, salt, and min/dill in a medium-sized bowl. Set this aside so that it stays at room temperature.
Make the butter sauce: Melt the butter in a small pan over medium heat. Once the butter starts to bubble add in red pepper flakes along with the paprika and stir them into the melted butter. Turn off the heat and let the peppers infuse into the butter while you poach eggs.
Poach the eggs: Crack each egg into small individual bowls and set aside. Then fill a wide walled pan ⅔ with water and place over high heat. Let the water come to a boil, then add the vinegar. Reduce the heat so that the water is barely at a simmer. Then gently slip each egg out of their bowls and into the water one at a time. Take the pot off of the heat and let each egg cook for roughly 4 to 5 minutes in the hot water or until the whites are firm yet the yolk is still runny. Remove the poached eggs from the water with a slotted spoon and place them on a plate lined with paper towels.
Assemble the çilbir by spreading the yogurt sauce on a plate. Top the yogurt sauce with the poached eggs and drizzle on the butter sauce. Serve immediately and enjoy!