Place dried mushrooms in a large bowl and pour about 1 cup of cold water over them (or more to cover). Cover the bowl and set it aside in a dry and cool place for 12 hours, preferably overnight. The mushrooms will absorb some of the water and soften.
Drain by pouring the water used to soak the mushrooms into a jug or bowl. Do not pour it out, we will use it to cook sauerkraut later on.
Slice the drained mushrooms into thin strips and then chop them finely.
In a skillet or large frying pan, heat up cooking oil, then add chopped onion and sauté until translucent.
Drain sauerkraut on a strainer, then rinse well with cold water and squeeze. Place in a large pot along with chopped porcini mushrooms, sauteed onion, shredded carrot, bay leaves, allspice berries, sugar, pepper, reserved mushroom soaking water and about 1 cup of additional cold water.
Mix everything thoroughly with a wooden spoon. Cover the pot with a lid and set the heat to low. Cook for about an hour, stirring well every 15 minutes. After an hour, remove the lid and sauté the filling for additional 20 minutes. Sauerkraut should be brown and very soft at this point. Taste and adjust with salt, pepper and sugar as needed.
Once ready, transfer to a large chopping board and chop finely. Set aside to cool down.
When your filling is nearly done, you can start preparing pierogi dough.
In a large mixing bowl, combine the flour and salt.
Into a jug pour boiling water and add cold butter. Mix until butter melts.
Gradually add water while stirring until the dough starts to come together. Then add beaten eggs and continue stirring.
Transfer the dough to lightly floured surface and knead for about 4-5 minutes until it becomes smooth and elastic (you can also do it in a stand mixer if you prefer).
Cover it with bowl or a damp cloth and let it rest for 30 minutes.
Divide the dough into 4 equal pieces. Roll each piece on very lightly floured surface until it is about ⅛ inch thick. Try not to add more flour as the pierogi will be hard to seal.
Use a round cutter (9cm/3.5-inch diameter) or a glass to cut out circles from the dough.
Place a small amount (about a teaspoon) of the sauerkraut and mushroom filling in the center of each dough circle.
Fold the dough over the filling to form a half-moon shape, pressing the edges to seal. You can use a fork to crimp the edges for a decorative touch.
Bring a large pot of salted water to a boil. Carefully drop the pierogi (about 12-15 pierogi at a time) into the boiling water and cook until they float to the surface. This usually takes 3-5 minutes. Remove the pierogi with a slotted spoon and transfer them to a plate.
In a separate pan, heat oil over medium heat and sautee chopped onion until golden. Pour sauteed onion over boiled pierogi and enjoy. Alternatively, add boiled pierogi to the pan with sauteed onion and fry until golden brown for a few minutes.
