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.
Make the filling: Place potatoes in a large pot, and cover with cold water. Season with salt. Bring to a boil; cook until fork-tender, 25 to 30 minutes. Drain; pass through a ricer. Stir cream cheese and butter. Season with salt and pepper.
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. Cover with plastic wrap to prevent dough from drying. Repeat with remaining dough.
Bring a large pot of salted water to a boil.
Fill pierogi: Place a 1 1/2-inch oval (about 1 tablespoon plus 2 teaspoons) filling in center of each dough circle.
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.
Make brown butter: Melt butter in a small saucepan over medium heat, swirling occasionally, until dark golden brown, 8 to 10 minutes.
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. Coat a platter with half the brown butter. Transfer pierogi to platter using a slotted spoon. Drizzle tops with remaining butter, and season with salt.