Heat butter with water in a pan over medium-high heat until it melts.
Add butter gradually to the flour mixture, then add the egg.
Knead dough for 5 to 7 minutes on a floured surface.
Cover the dough and let sit for 30 minutes.
Boil potatoes until a fork slides through easily, about 25 minutes.
At the same time, fry onions with the butter over medium-high.
Mash the potatoes and combine with sour cream, cheddar, and onions (but reserve some onions for serving). Season with salt and pepper.
Roll out dough. Cut out 3-inch circles using a cookie cutter or a glass. Roll each circle slightly bigger and then add a heaping teaspoon of filling, pinching to close.
Boil pierogies until they float, and then for a further 2 minutes.
Serve with reserved onions and extra sour cream.
Notes
Freeze pierogies before boiling if you won’t eat them right away. Freeze on a baking tray until solid, then transfer to freezer bags or containers. When boiling from frozen, give them an extra minute.
You can pan fry the pierogies in butter after boiling if you want.