Slice the peeled potatoes into 1/2-inch-thick rounds. Place them in a medium saucepan, cover with water, and set over medium-high heat. When it comes to a boil, continue cooking for about 15 minutes or until potatoes are very tender. Remove from heat and drain potatoes.
Add cheese, butter, and salt to saucepan and either mash by hand or use a hand-held mixer to blend until smooth and fluffy. Transfer potato mixture into a small bowl or baking dish, cover, and refrigerate (the potato filling is easier to work with when it's cooled).
In a medium mixing bowl, combine bread flour and salt. Make a well in the center and add eggs. Mix gently, adding water as needed, until the dough comes together (you may not use the full amount of water, I usually have a few tablespoons left).
Knead by hand, either in the bowl or on a lightly floured surface, just until the dough forms a firm ball, about 30-60 seconds. Lightly coat the ball in flour and cover (to prevent dough from drying out) and set aside for about an hour.
Line a baking sheet with parchment paper or a silicone baking mat and set aside.
When dough is done resting, cut it in half. Keep one half covered in the bowl. If you have a pasta machine, feel free to use it to roll out the dough. Otherwise, turn out the other half onto a lightly floured work surface. If it's sticky, sprinkle a little bit of flour over it and work it in gently. Roll out the dough to about 1/8-inch thickness. (If the dough springs back a lot, allow it to rest about 10 minutes and try again.) Cut rounds in the dough with a 3-inch biscuit cutter or the top of a glass or jar.
Remove filling from the refrigerator. Using a spoon or tablespoon-sized cookie scoop, place a scant tablespoon of filling (a bit less than a tablespoon) into the center of each dough round. Gently fold edges together, pinching with your fingers to achieve a good seal. You may need to gently stretch the dough to get it up and over the filling to pinch together. Avoid getting filling in the seal or your pierogi may leak while cooking!
Placed finished pierogi on the lined baking sheet. Re-roll your scraps, allowing your dough to rest if it's difficult to roll. Repeat steps with other half of dough. You should get about two dozen pierogi. (You will have filling left--eat it with a spoon or save it for later!)
Freeze baking sheet of pierogi until firm and then transfer into a freezer bag or container until ready to use.
To cook, set a large saucepan of well-salted water over high heat and bring to a boil. Meanwhile, melt about a half-stick of butter in a skillet and add 1/2 onion, diced or sliced (your preference), and cook until onions are browned. Carefully drop pierogis into boiling water (a half-batch at a time) and boil for 5 minutes.
When all pierogis are boiled and buttered, serve!
Frozen pierogis can be kept for several months. Cover and refrigerate any leftover cooked pierogi and eat within 3-4 days.