Melt a heaping tablespoon of butter in a pot and add a bit of oil so that the entire bottom is covered with fat.
Sprinkle 2-3 tablespoons of all-purpose flour over the butter, to make a rough. Keep stirring and sautéing until it starts to smell nutty and the mixture lightly browns.
Pour hot broth into the roux and whisk everything together.
Melt 1 tbs of butter in a frying pan and sauté the mushrooms until wilted and lightly browned
Add sliced potatoes, sauteed mushrooms, a bay leaf, and allspice to the hot broth & rough mixture.
Cook until the potatoes are soft, which takes about 10–15 minutes.
Remove the bay leaf and allspice. Pour in heavy whipping cream. Season with salt and pepper, add chopped fresh dill, and balance the flavor with sugar and vinegar. The ratio of sugar to vinegar is true alchemy; I use about 2 tablespoons of vinegar and 2 tablespoons of sugar, but it really depends on your taste, so start with less and adjust as needed.
Lastly, whisk in the eggs; alternatively, you can also serve them poached.