FOR THE FILLING, cook potatoes in boiling salted water until tender, and drain. If you have a ricer, rice potatoes and stir in butter while hot, otherwise mash with electric beaters. Season to taste and add nutmeg. If adding cheese, allow potatoes to cool, then stir in grated cheese. Chill potato filling before using.
FOR THE DOUGH, combine flour and salt. In a separate bowl, whisk eggs, milk and water. Add liquid to flour and stir until blended. The dough should be soft enough that it sticks to the bottom of the bowl, but not so soft that it doesn’t hold its shape. You can always add a little more liquid or flour if you need to. Cover dough and let rest 1 hour.
On a lightly floured surface, roll out dough as thinly as possible without tearing. Use a round cutter 3 inches in diameter and cut out circles of dough. Place 1 tablespoons of potato filling into the center of each perogy and pinch the perogy edges closed. Perogies can be frozen in one layer on a baking sheet or kept chilled until ready to cook.
To serve, melt butter in a sauté pan over medium high heat. Add onions, stirring constantly until onions are richly browned. Reduce heat to medium low.
TO BOIL VARENYKY: In a pot of boiling salted water, cook about half of the perogies until they float, about 5 minutes. Remove from water with a slotted spoon, add to browned butter and gently stir with rubber spatula or wooden spoon to coat them, cook remaining perogies and stir gently into browned butter. Makes 24 to 30 perogies.