Make the dough: Whisk together egg and sour cream. Whisk in milk and water. Stir in flour, 1 cup at a time.
Turn out dough onto a floured surface. (Dough will be loose and sticky.) Using a bench scraper, turn and fold dough to knead, dusting with flour as needed, until elastic and no longer sticky, 8 to 10 minutes. (Dough will come together as you knead it. Be careful not to add too much flour, since it will toughen the dough.) Cover with an inverted bowl; let rest for 1 hour.
Divide dough into 4 equal pieces. Line a rimmed baking sheet with a clean linen towel, and dust generously
with cornmeal to prevent sticking.
Roll out 1 piece of dough on a lightly floured surface into a 1/8-inch-thick round (keep other pieces covered).
Cut out circles very close together, using a 3-inch cutter or glass (5 inches for the plum version). Cover with plastic
wrap to prevent dough from drying. Repeat with remaining dough.
Fill pierogi: Place filling in center of each dough circle. Bring a large pot of salted water to a boil.
Holding 1 circle in your hand, fold dough over filling. Pinch edges, forming a well-sealed crescent.
Transfer to cornmeal-dusted towel, and loosely cover with plastic wrap. Repeat with remaining dough circles and filling. Working in batches, transfer pierogi to boiling water. They will sink to the bottom and then rise. Once they have risen, cook through, about 2 minutes more (or up to 4 minutes for plum pierogi, depending on ripeness). For savory pierogi, coat a platter with half the butter. Transfer pierogi to platter using a slotted spoon. Drizzle tops with remaining butter, and season with salt. For sweet pierogi, transfer to a platter using a slotted spoon, and dot with sour cream.