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1 For the dough, whisk the egg in a medium bowl; whisk in the sour cream until smooth.
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Add the milk and water; whisk until combined.
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Slowly add about 3 1/2 cups of the flour, stirring to combine.
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2 Turn the dough out onto a well-floured surface; knead, working in about 1 cup of the flour, using a scraper if necessary.
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Continue kneading 8-10 minutes, working in about 1/2 cup of the flour, just until dough is elastic and no longer sticky.
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(Don't add too much flour, as this will toughen dough.)
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Place in a lightly floured bowl; cover with plastic wrap.
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3 For the filling, place potatoes in a large saucepan; cover with cold, salted water.
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Heat to a boil over high heat; cook until fork-tender, about 20 minutes.
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Drain; mash potatoes.
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Add half of the melted butter, the Cheddar and cream cheese; mash until well incorporated.
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Season with 2 teaspoons of the salt and pepper to taste.
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4 Roll out dough to about 1/8-inch thick on a floured surface.
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Cut out circles with 2 1/2-inch glass or a cookie cutter.
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Gather dough scraps; roll out again.
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Continue cutting.
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5 Place a clean towel on work surface.
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Place about 1 1/2 tablespoons of the filling in the center of each dough circle.
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Holding a circle in your hand, fold dough over filling; pinch the edges, forming a well-sealed crescent.
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Transfer pierogi to towel.
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Repeat until all dough circles are filled.
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(You may have filling left over; refrigerate for another use.
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).
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6 Heat a large stockpot of water to a boil over high heat; add pierogi in small batches.
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Once they float to the top of the pot, cook about 1 minute.
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Meanwhile, in a small saucepan, cook the remaining butter over medium-high heat until it is nut-brown in color, about 6 minutes.
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Pour butter onto a serving platter, leaving any burned sediment behind.
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Transfer pierogi from pan with a slotted spoon to the platter.
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Serve hot.
