Bring a large pot of heavily salted water to a boil. Cover it so it doesn't evaporate too much while you make the dumplings. Line a sheet pan with a kitchen towel and lightly dust it with flour. Have another kitchen towel ready to cover the dumplings once shaped.
Use a bench scraper or large knife to cut the dough ball into 4 pieces. Set one aside and cover the remaining 3 with the plastic. Initially, it might feel really sticky, but that will dissipate almost immediately once you start to roll it.
On a lightly floured surface, roll out the dough until it is between ⅛ and 1/16 of an inch thick. Keep turning the dough as you roll it out to keep it from sticking as you would pie dough.
Use a round cookie cutter about 3 inches in diameter (or a glass) to cut circles into the dough. Get them as slose together as possible to get the maximum number of dumplings per piece of dough.
Place some potato filling in the center of each dumpling wrapper (about 1¼ tablespoons for this size circle). I use a cookie scoop to ensure that I don't everfill or underfill them.
To shape the varenyky, gently pick up a round of dough with potato on it. Use your thumb to gently press down a bit on the filling, while using your other hand to pinch both sides of the circle together in the middle of the dumpling. (Sometimes it's hard to get the dough to stick when you first bring it together. Just hold your fingers in place for a second or two to warm it up and it should stick more easily.)
Pinch the varenyky closed, working your way toward your thumb that's on top of the filling and making sure there are no air pockets. Then pinch the dough closed in the other direction.
Gently double back and pinch the seam from one side to the other to make sure it is closed. I kind of move my thumb forward while moving my index finger backward which closes the seam and makes a nice pretty edge.
Place the shaped varenyky on the lightly floured towel and cover with another kitchen towel to keep them from drying out while you shape the rest. Make sure they are not touching each other. Roll the scraps into a ball and cover them with plastic.
Continue with the next 3 pieces of dough, one at a time, rolling, cutting, filling, then placing them on the sheet pan under the kitchen towel. Finally, combine all the scraps and roll them out, cut, fill and shape. Do not roll out the final scraps. The dough will be too tough. But you can cut them into strips and boil them as noodles if you like.