To make the yogurt sauce, whisk together the yogurt, garlic, lemon juice, salt, and pepper in a bowl, then cover and set aside. I intentionally do this first so that it can come to room temperature while I poach the eggs.
To poach the eggs, fill a medium saucepan with water (it should be at least 4 inches deep) and bring it to a boil over medium-high heat. Once it comes to a boil, turn the heat down to low and let it come to a simmer. Stir in the vinegar.
While you are waiting for the water to come to a boil, place a fine mesh strainer over a small bowl and crack an egg while being super gentle to make sure not to disturb the yolk. Let the excess loose (watery) egg whites drop through and then transfer into a ramekin. Repeat the same process for the rest of the eggs.
Place a sheet of paper towel on a large plate and set it aside.
When ready to poach the eggs, stir the simmering water to create a vortex in the middle, carefully drop the egg in the middle, and cook for 3 to 3 ½ minutes until the whites are firm and the yolk is set but still soft and runny. For a medium yolk, cook 4 minutes. If you prefer the yolks to be fully set, cook for 5 minutes.
Carefully remove the egg with a slotted spoon, then transfer it to the paper-towel-lined plate. Repeat the process with the remaining eggs*.
Right before serving, melt the butter in a small skillet over medium-heat. Add the Urfa chili pepper (or any red pepper flakes) and salt. Let it cook, stirring gently with a wooden spoon, for 15-20 seconds until fragrant, lightly foamy, and a reddish color from the pepper, but not browned.
To assemble: Using the back of a large spoon, spread half of the yogurt sauce on a plate. Add two poached eggs on top and drizzle them with the sizzling chili butter sauce.
Garnish with fresh parsley or chopped dill; sprinkle it with a pinch of salt and pepper to taste. Repeat the same process for the second serving. Serve immediately while still warm.