Begin by peeling and cutting your potatoes into even chunks. Place them in a pot, cover with cold water, and add a pinch of salt. Bring to a boil and simmer until fork-tender, about 15-20 minutes. Drain and keep warm.

In a separate pot, fill with water and heat to a gentle simmer at around 65°C. Crack the eggs into a small bowl, then gently slide them into the water. Cook for 5-6 minutes, ensuring whites are set and yolks remain soft. Remove with a slotted spoon and drain on paper towel.

Pour cream into a small saucepan over low heat and warm gently. Stir in freshly chopped dill and let it infuse. As the sauce thickens, season with salt, pepper, and a touch of brown sugar. Remove from heat just before serving.

On each plate, create a cozy bed with warm potatoes. Place poached eggs atop and generously spoon the dill sauce over them.

Garnish with additional fresh dill for color and flavor. Ensure everything is warmed through and check seasoning before serving warm.