Place the potatoes in a large pot and cover with water. Bring to a boil and then simmer until fork tender, about 15 minutes. Drain the water and mash the potatoes. Place the mashed potatoes in a mixing bowl and chill in the refrigerator until cold.

In a small skillet over medium heat, warm the oil and then saute the onion until tender. Let them cool for a few minutes and then add the cooked onions to the bowl of cold mashed potatoes. Place the sauerkraut in a strainer and rinse very well under running water, for at least 3 minutes. Drain thoroughly and add the sauerkraut to the bowl with the potatoes. Add the cheese and stir to combine. Store in the refrigerator until ready to make the pierogies. The filling can be made a day or two in advance and refrigerated until needed.

Place the flour and salt in a large bowl and stir to combine. Make a small well with your hand and crack the egg into it. Stir to combine and then gradually add the hot water, mixing with your hands until the dough is very sticky and well combined. If it becomes too sticky and wet, add just a tablespoon or so more flour. Chill the dough in the refrigerator for at least an hour.

Bring a pot of water to a boil. Sprinkle a large board with flour. Transfer the dough to the floured surface and flip it over a few times to coat with flour. Gently knead the dough just a few times with your hands, adding a sprinkling of flour as needed.

To Shape Pierogies: Divide the dough into four sections and roll out one of the sections very thin, to about 1/8" thickness. Cut into circles and place 1 tablespoon of the potato filling on one side. Fold over the circle and pinch around the dough. Place the prepared pierogies in a single layer onto a waiting plate. Don't stack them or the dough will stick together.

To Boil Pierogies: Drop the waiting pierogies into the water, a few at a time. Use a spoon or spatula to gently nudge them off the bottom of the pot if they stick. When they float to the surface of the water, lift them out carefully with a slotted spoon. Place them on a buttered plate and drizzle a small amount of melted butter on top of them to prevent sticking together. Let the boiled pierogies cool for a few minutes before frying them.

To Fry Pierogies: In a large skillet over medium heat, melt a tablespoon of butter. When the butter sizzles, place the boiled pierogies in the skillet in a single layer. Let them cook 2-3 minutes, until lightly browned on the edges, flip over and repeat. The pierogies will not turn brown all over, but the edges should crisp and be light brown. Remove the finished pierogies to a buttered baking dish.