Despite the international diversity that makes up the East Village's Eastern European population—Ukrainian, Polish, and Russian, and those are just the big three—we tend to refer to their dumplings as one in the same: pierogi.
Pierogi,* the Polish term for filled dumplings of potatoes, sweet or savory cheese, vegetables, or meat, have subtle differences from ear-shaped Russian pelmeni and mild Ukrainian vareniky, the last of which you'll also find in Russia, and heaven help you if you confuse them with pelmeni. That said, they're all similar: sometimes boiled, sometimes fried, with moderate to thick wrappers of flour, water, and sometimes egg, all with similar rosters of fillings. But there is another kind of dumpling all its own.
* That's the plural, by the way; the singular is pieróg.
I'm talking about halusky, which have more in common with spaetzle or gnocchi than these other dumplings. Halusky are solid, unfilled dumplings made from a dough of flour and water, sometimes with egg, sometimes with mashed potato. Depending on how they're formed they can take on the appearance of log-like dumplings, as in the photo above, or more spaetzle-like noodles. The East Village's Eastern European restaurants prefer the dumpling-like style.
The name "halusky" refers both to the dumplings and a dish typically made from them: a plate of the dumplings topped fried cabbage or sauerkraut and meat like bacon or sausage. You'll find this dish at Ukrainian East Village Restaurant, where a sizable portion of halusky comes with caramelized sauerkraut and bacon ($10.50).
Truth be told, I don't love the cooking at this restaurant, which leans towards the bland and greasy. But it's a vibrant community center with cute decor and, if you're lucky, tango going on in the back room. If you need a hideaway from the bustle of the city—literally, in this case, as you walk down a long hallway to enter the restaurant—there are worse places to do so, especially if comfort food with a Brandy Alexander is on your mind.
So some advice: load up on the restaurant's quality sour cream and spread it liberally over your plate of dumplings. Don't shy on the salt as well—the bland dumplings need it. You'll be rewarded with a homey, carby, and pleasantly oily plate that'll add one more mark towards your Dumpling Master merit badge.