* Rinse kraut and squeeze thoroughly dry.
Dumplings: In a food processor, blend the flour and 1/2 teaspoon salt. With the motor running, add the egg yolks and the oil through the feed tube, then pour in the water, in a slow, steady stream, until the dough forms a ball around the blade. Transfer the dough to a floured surface and knead until smooth, about two minutes. Cover with a linen or cotton (not terry) kitchen towel and let stand half an hour.
Divide the dough in half and shape into two balls. Keep one ball covered with the towel. On a floured surface with a floured rolling pin, roll out the dough to a very thin sheet, about 1/16" thick, making sure it doesn't tear. With a round cookie cutter, cut out circles about 3" in diameter. Gather the scraps together into a ball and set aside, covered.
Have a bowl with the egg white by you. Place a heaping teaspoon of the desired filling in the middle of each circle. Fold the dough over the filling to form a semi-circle. Brush the edges with the egg white and press the edges firmly together with the tines of a fork to seal. Place the vareniki as they are made on a lightly floured large baking sheet about 1" apart and keep covered with a damp tea towel. When finished with the first batch, roll out the second ball of dough and make a second batch. Add the leftover scraps of dough to the scraps left from the first batch, knead into a ball, and roll out for a final batch of varenyky.
Meanwhile, in a large pot, bring 6 quarts of salted water to a boil. Reduce the heat to medium so the water simmers and carefully lower half the varenyky into the water. Boil, stirring occasionally with a wooden spoon to prevent sticking, until they rise to the surface and are cooked through, six to seven minutes. With a slotted spoon, carefully remove the varenyky to a colander and drain thoroughly. Transfer to a deep serving bowl and toss with half the butter. Repeat for the second half of the varenyky.
Potato filling: Melt the butter in a small skilled over medium heat. Add the onion and saute, stirring occasionally, until nicely browned, about 15 minutes. Remvoe from heat and cool slightly. In a large bowl, combine the potatoes and cheese. Add the sauteed onion along with the melted butter and mix well. Season with salt and pepper and use for filling the varenyky.
Sauerkraut filling: Saute the bacon in a large skillet over medium heat until it renders its fat. Remove the bacon and reserve. Drain off all but two tablespoons fat. Add the onion to the skillet and saute, stirring frequentlly over medium heat until nicely browned, almost 15 minutes. Turn the heat up to medium-high, add the kraut, and saute, stirring, until softened and cooked throughout, 10 to 15 minutes. Stir in the tomato paste, sugar, and broth. Reduce the heat to low, then cover and simmer for 20 minutes. Remove from the heat and cool to room temperature before using to fill the varenyky. Use the reserved bacon as topping.