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1. To make wrappers, combine egg yolk, milk, oil, and water in a small bowl. Place flour in a large bowl, pour in wet ingredients, and fold together to form dough.
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2. Transfer dough to a lightly floured board and knead for 3 minutes. If too sticky, add small amounts of flour. When dough is smooth, form into a ball, transfer to a small bowl, cover with plastic wrap, and refrigerate for 20 minutes.
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3. To make filling, melt butter in a large pan, add onions, and saute over medium heat, stirring occasionally, until browned, about 10 minutes.
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4. In a large bowl, combine 3/4 cup of cooked onions, potatoes, cheese, salt, and pepper. Set aside at room temperature. Reserve remaining onions for topping.
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5. In a small bowl, combine egg whites and water. Set aside for assembly.
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6. Remove dough from refrigerator and divide into three sections. Place one section on a well-floured work surface and roll out until very thin (about 1/16 inch). Cut circles of dough using a round cookie cutter (or a jar lid or juice glass).
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7. Place a heaping spoonful of the filling in the center of each circle, leaving a border. Brush some egg white mixture on half the outer border and fold dough over into a half-moon shape. Crimp edges with a fork.
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8. Transfer pierogi to a floured board or platter. Do not stack. Repeat with remaining dough and filling.
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9. Cook pierogi in batches for 4 minutes in a large stockpot of salted boiling water. Drain and transfer to a serving platter. Repeat with remaining pierogi, allowing water to return to a full boil each time.
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10. Serve with the reserved sauteed onion and oodles of sour cream or applesauce.
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Try them fried: Cook in boiling water as above for 2 minutes rather than 4, then saute in butter until golden brown.
