In a medium bowl, whisk the egg. Add the sour cream, and whisk until smooth. Add the milk and 1 cup water, and whisk until combined. Slowly add about 3 cups flour, and stir with a wooden spoon to combine.
Turn dough out onto a well-floured surface, and work in about 1 cup flour as you knead. Use a plastic scraper to lift the dough as it will stick to the counter before the flour is worked in. Continue kneading for about 8 to 10 minutes, working in another 1/2 cup flour. The dough should be elastic and no longer sticky. Be careful not to add too much flour as this will toughen the dough. Place dough in a lightly floured bowl, and cover with plastic wrap; set aside while you prepare the filling.
Cook potatoes in salted boiling water until fork-tender. Drain, and mash with a potato masher. Add melted butter and cheeses, and continue to mash until well-incorporated. Season with salt and pepper to taste. Place a large pot of salted water over high heat, and bring to a boil. Lay a clean linen towel on your counter, and evenly distribute cornmeal on it to prevent sticking.
On a floured surface, roll out dough to about 1/8-inch thickness. Using a 2 1/2-inch-diameter glass or cookie cutter, cut out as many circles as possible. Gather dough scraps together, roll out again, and continue cutting.
Form filling into 1 1/2-inch balls, and place a ball in the center of each dough circle. Holding a circle in your hand, fold dough over filling, and pinch the edges, forming a thoroughly sealed crescent. Transfer to linen towel. Continue until all dough circles are filled. Add pierogi to the boiling water in batches. They will sink to the bottom of the pot, then rise to the top. Once they rise, let them cook for about 1 minute more. Meanwhile, drizzle platter with melted butter. Remove pierogi from pot, and transfer to platter to prevent sticking. Serve immediately.
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