Using a 5-quart standing mixer with a dough hook, add the flour, eggs, salt, and some of the water to the bowl. Mix for 1 minute on medium speed until the dough begins to come together. Then add the butter. If the dough is still dry, add more water; if it is too wet, add a bit more flour. Increase the speed to high for 3 minutes. The dough should now be formed; it should not be sticky to the touch.
Rest the dough for 30 to 45 minutes, covered with a moist towel or plastic wrap. You can also wrap the dough and refrigerate it until the next day; just remember to bring the dough to room temperature before you use it.
Place a large stockpot filled with water on the stove and bring it to a boil.
Cut the dough in half. On a dusted countertop, begin to roll out your pierogi dough. Roll it fairly thin, to about ⅛" thick, and cut circles out with a 4" round cutter. Be sure to keep the circles close together to get the most yield out of the dough. Babcia would never reuse the dough (poor form). Repeat with the other half of the dough.
Hold the circle in your hand and fill it with as many blueberries as the dough can comfortably hold. Then cup the pierog (singular) like a taco and begin to pinch the edges to seal. Continue until you have them all filled.
To cook the pierogi, carefully drop them one at a time into the boiling water (you can probably cook 4 to 6 at a time). Cook for 2 minutes per side. Once they are done shock them in an ice bath. Drain and set aside.
You can either enjoy the goodness now or you can refrigerate them. They last about 5 days, but they are delicious, quick portable snacks.
Serve the pierogi with sugar, hot butter, and sour cream. Or just eat them as they are.