Start by making the filling. In a saucepan heat up the oil, add the onion and cook over a low/medium heat for a couple of minutes until softened, stirring often. Add the sauerkraut, mushrooms, seasoning (mainly pepper at this point), stir, cover and simmer for about 40-45 minutes, stirring often (do not let the mixture burn). You may have to add a drop of water if the mixture becomes too dry and starts sticking to the pot. Remove from the heat and puree 1/3 of the mixture. (ensure you puree the porcini but the mixture should not be completely smooth). Combine with the remaining sauerkraut. Add the breadcrumbs, stir, adjust the seasoning if necessary (the mixture should be tangy, sharp and peppery) and set aside to cool.
While the sauerkraut filling is cooking prepare the dough. Tip the flour onto your work surface, add the salt, make a well in the middle, add the butter and start combining using a large knife, cutting through the mixture and gradually adding the water into the middle and gathering up the mixture with the knife to prevent the water from escaping. When all the water has been added and the dough comes together continue kneading for 6 more minutes until smooth and soft. If the dough sticks to your hands add 1 teaspoon of flour and knead in. Cover the dough with a bowl and set aside for 20 minutes.
Divide the dough into 2 parts (easier to work with) and roll out one part (keeping the other covered) on a lightly floured surface as thinly as possible, 2-3 mm in thickness (don't worry, the dough is quite elastic and won't tear easily). Using a glass with a diameter of about 7 cm make round shapes (gather up the excess dough and add to the other dough part).
Place a small amount of the filling (about a teaspoon) in the middle of the round dough shape (if the dough is sticking to the work surface use a knife to help it come off but take care not to tear it). With tips of your fingers stick the edges of the dumpling together - go over the edge twice for each dumpling (your fingertips should be dry so it's a good idea to keep a sheet of paper towel near you to wipe them if they get sticky).
Place the dumplings on a lightly floured surface and cover with a tea towel.
Fill a large pot with salted water, add 1 teaspoon of oil, cover and bring to boil. Continue making the dumplings as you are waiting for the water to boil. When the water starts boiling carefully place the dumplings in the pot one by one (up to 15 per batch) and quickly but carefully stir with a wooden spoon. When the dumplings come up to the surface cook them for about 4 more minutes, then switch off the heat and using a slotted spoon transfer them onto a large plate. Drizzle with a little olive oil/butter and serve! Yields 35-40 pierogi.
The sauerkraut filling can be made ahead so that you'll have one thing less to do when putting the recipe together. Make the filling, allow it to cool completely, then refrigerate it, covered, overnight. Give the mixture a stir before making the pierogi.
To wash the porcini mushrooms place them in a small bowl, add a bit of water and rub with your fingertips to remove any grit. Rinse under cold water and add into the sauerkraut.
The filling might occasionally stick to your fingers as you are making the dumplings. It is a good idea to wipe your fingertips (paper towel works well) and dust them with flour from time to time. The pierogi will look neater and will be easier to make.
Try to roll out the dough as thinly as possible (approx. 2-3 mm thick). Don't worry, it won't tear easily.
You can make the pierogi dough using either butter or oil. Use either vegetable or olive oil and add 2.5 tablespoons into the dough.
Storing and reheating: If you aren't using the dumplings straight away place them in an oven proof dish (the oil will prevent them from sticking to one another), allow to cool and refrigerate (for up to 3 days) until they are ready to be reheated (in the same dish). Cover with tin foil and reheat in the oven (180-190 C). You can also reheat individual portions in the microwave.