Before you make the dough, put a large pot of salted water on to boil for cooking the pierogi, so it is ready when you are. I use an 8-quart pot filled about halfway.
In a large bowl, combine the flour and salt.
In a small bowl or measuring cup, whisk together the egg and water.
Add the egg mixture to the flour mixture. Stir until the dough comes together in a smooth, not-very-sticky ball (you can use your hands if you'd like or find it easier). Be careful not to over-work the dough, as that can make it tough. Note: If it is too dry and is not coming together, add water, a little bit at a time, until it does. If it is too wet and sticky, add flour, a little bit at a time.
Cover the bowl with a damp clean towel and let the dough rest for 30 minutes (or you can wrap the dough in plastic wrap).
Divide the dough in half. Place one half of the dough on a lightly floured surface. Lightly sprinkle the top of the dough with flour so the rolling pin doesn't stick. Keep the other half of the dough in the bowl covered with the damp towel (or wrapped in plastic wrap) so it doesn't dry out.
Roll the dough out to ⅛-inch thickness. Tip: You may find this easier if you roll it out about halfway, lightly flour it, flip it over, and then roll it out the rest of the way.
Using a 3½-inch round cutter, cut the dough into as many circles as possible. Gather the scraps into a ball and put them in the covered bowl (or wrap with plastic wrap) so they don't dry out in case you need to use them.
Place about 1 tablespoon of the cooled potato and cheese filling on one half of each dough circle. Tip: put it on the side of the dough that has the least amount of flour - this will make the pierogi easier to seal.
Fold the dough over the filling, stretching it a little if needed, and pinch the edges to seal. Pinch tightly and make sure they are sealed very well so they don't open up when boiling. Place the pierogi on a lightly floured surface so they don't stick until ready to boil.
Repeat with the remaining dough and filling, rolling out the dough scraps if you need to. Note: the scraps might be a little harder to roll since the dough was already rolled once.
About a dozen at a time, place the pierogi in the boiling water. Give them a gentle, quick stir so they don't stick to each other or the bottom of the pot, and then cook until they float to the top. Once they float to the top, let them cook for another minute.
Using a slotted spoon, remove the pierogi from the water and place them on a wire rack to cool and dry (or you can eat them immediately if you aren't going to fry them). Once cooled, freeze any pierogi you don't plan to eat immediately/today (see notes below).
To fry a dozen pierogi, melt 1 tablespoon of butter and 1 tablespoon of vegetable oil in a large skillet over medium heat (I use an 11 to 12-inch skillet). Add the pierogi in a single layer and cook until lightly golden on both sides. Season with salt and pepper to taste.
If desired, serve with sauteed onions.