When the fillings are completely chilled, it's time to set up your work spaces. It's best to have three separate areas: 1. the dough area; 2. the pierogi making area; and 3. the pierogi cooking area.
For the dough area, clear a very large space to roll out the dough. Heavily dust it with flour and make sure to have your rolling pin, dough round cutter, and extra flour at-the-ready.
For the pierogi shaping area, lay down a tablecloth or thin sheet of some kind, also dusted with flour, to keep the pierogi from sticking. Also put out the filling, a spoon, a ravioli press or cup, and extra flour.
For the last space, you'll need a large pot filled ¾ of the way with water, a baking tray with softened butter (measured with your heart), a spider or slotted spoon, and a separate tray or serving dish to place the boiled and buttered pierogi.
Now you are ready to make the dough. To do this, use a stand mixer, with whisk attachment, to whip the eggs until they start to foam and gain volume. Add the salt and continue whisking for about 10 seconds. Then, add flour and water to the mixer a little at a time. Use all of the water but not all of the flour. The dough should be the consistency of pancake batter. At this point, switch out the whisk for the dough hook and add in the rest of the flour. The dough is ready when the flour is fully incorporated and the dough pulls away from the sides of the bowl. If the dough is sticking at all to the bottom of the bowl when fully mixed and all the ingredients are in, add about ½tbsp (4g) of flour at a time until the dough is spinning freely of the bowl.
As soon as this is achieved, stop the mixer and let the dough hydrate under a damp cloth for 20 minutes. When rested, scrape the dough out of the bowl onto your floured work surface. Cut a portion of the dough off and place the damp towel back over the remaining dough. Roll your portioned dough out as thin as you can get it (roughly 1/16in or .16cm). Add flour to your surface, as you roll, to stop the dough from sticking to your counter. When ready, cut 3.5in (9cm) circles out of the dough with a large biscuit or cookie cutter. Make each new circle as close as possible to the previous circle. Once all the circles have been cut into your rolled out dough, pull up the scraps and ball them up. You can throw this away or also put it under the damp towel to rehydrate and relax.
Next, turn on the burner for the pot full of water, so it comes to a slow boil while you make the pierogi.
Now, use a tablespoon or #50 cookie scoop to portion out some filling. There should be about 20-25g of potato filling by weight per scoop and/or 15g of sauerkraut per scoop. Make sure your hands are well floured. This helps you shape the filling into a semi-circle without it sticking to your fingers. Place your shaped filling onto a dough round. You want it to be in the center of the back-half of the circle, not the exact center. This way, it's easier fold the front-side of the dough over the filling without it stretching and breaking.
As soon as you have the front of the circle folded over the filling, press down where the dough overlaps to create an initial seal and keep the filling from falling out. (This is more important with the sauerkraut filling, but it's still a good habit to have.) Next, use a round ravioli stamp or a plastic cup to cut off the excess edge of the pierogi and make a really good seal. (For a more traditional style, use your fingers to press the edges together) Make absolutely sure to inspect the edge for holes. If they aren't sealed, water can get in during the boiling process.
When about 8-10 pierogi are made, place them into the pot to gently boil. After about 6-8 minutes, take out the pierogi and transfer them to the baking tray with several pats of butter. Shake the tray from side-to-side, ensuring that the pierogi are all very well coated in butter. Transfer the finished pierogi to a secondary dish or tray for storing or serving.
You can then begin again wherever needed: boil the next 8-10 pierogi; shape new pierogi; or roll out more dough, etc. until all of the filling and dough is gone.
Top with onion butter, salt, and/or sour cream to enjoy.