Spoon yogurt into a bowl. Press garlic in and mix to combine. Season with salt, pepper, cayenne and dill. Mix thoroughly. Set aside at room temperature.
Melt butter in a saucepan over medium heat; heat until bubbles begin. Add cumin, paprika, chili flakes and parsley. Stir until color is uniform, then turn off heat and let sit. Set aside.
Fill a large saucepan with 2 to 3 inches of water and bring to a boil. Reduce heat to medium-low, pour in vinegar, and keep water at a simmer. Crack an egg into a small bowl then gently pour egg into simmering water. Repeat with remaining eggs. Cook eggs until the whites are firm and the yolks have thickened but are not hard, 2 - 3 minutes. Remove eggs from water with a slotted spoon, dab with a paper towel to remove excess water, and set aside on a plate.
Dollop yogurt mixture onto serving plates. Use a spoon to spread yogurt out into a bed for the eggs, create dips in yogurt to catch the oil. Place eggs into yogurt and drizzle chili butter on top. Sprinkle sea salt on top, garnish with olive oil (optional) and serve with toasted bread.