Add the Greek yogurt, garlic, dill, and a pinch of salt to a bowl. Mix until well combined and set aside.

Add the butter to a small saucepan over medium-low heat. Cook, swirling the pan regularly, until the butter begins to foam and the milk solids turn lightly golden brown.

Remove the pan from the heat. Stir in the olive oil, chili flakes, and paprika, then set aside.

Bring a deep saucepan of water to a boil. Reduce the heat to maintain a gentle simmer, then add the vinegar.

Place a fine-mesh strainer over a small bowl and crack one egg into the strainer. Allow the loose, watery egg white to drain away, then transfer the egg to a small bowl.

Stir the simmering water to create a gentle whirlpool. Carefully slide the egg into the center, cover, and poach for 3 minutes.

Remove the egg with a slotted spoon and allow any excess water to drain. Repeat with the second egg.

Spread the yogurt over a serving plate or shallow bowl.

Place the poached eggs over the yogurt and season lightly with salt.

Spoon the warm chili butter over the eggs and yogurt. Finish with additional fresh dill and serve immediately.