These pierogi are lighter and more delicate than most versions.
Make the dough: Whisk together flour and salt in a medium bowl. Make a well in center, and add egg and milk. Using your hands, gently draw flour mixture into egg mixture. Continue to work dough with your hands or a spoon until it comes together to form a ball. (Add up to 2 teaspoons milk if dough is crumbly.) Place dough on a lightly floured work surface, and knead until smooth and slightly elastic, about 5 minutes. Transfer to a clean surface, and cover with an inverted bowl or wrap in plastic. Let dough rest for 1 hour.
Make the onions: Melt butter with the oil in a skillet over medium heat. Add onions, and season with salt. Cook, stirring frequently, until onions are soft and golden brown, 12 to 15 minutes. Remove a heaping 1/4 cup onion mixture, and finely chop; reserve for filling.
Make the filling: Place potatoes in a saucepan, and cover with water by about 2 inches. Add 1 teaspoon salt, and bring to a boil. Reduce heat, and simmer until tender, 17 to 20 minutes. Drain, and immediately return to pan. Place on stove (with burner off), and let dry for 10 minutes. Pass through a ricer or a food mill into a large bowl. Stir in stock, reserved chopped onions, remaining 1 teaspoon salt, and pepper to taste.
Make the dumplings: On a lightly floured work surface, roll out dough to a scant 1/8 inch thick. Cut out rounds with a 2 3/4-inch cutter. Place on a baking sheet, and cover with a clean kitchen towel.
Place 1 dough round on a work surface. Spoon 1 heaping teaspoon filling onto center of bottom half of round. Fold top half down over filling, gently pressing to squeeze out any air and seal. Stretch edges outward slightly to make them thinner. Return to baking sheet, and cover. Repeat. (Pierogi can be refrigerated overnight; wrap towel-covered baking sheet tightly with plastic.)
Bring a large pot of salted water to a boil. Add 8 dumplings. Cook, stirring gently and occasionally, until dumplings rise to the surface, about 4 minutes. Using a slotted spoon, transfer to a plate lined with paper towels, and gently pat dry. Transfer to a warmed serving platter, and cover to keep warm. Repeat with remaining dumplings.
Melt butter in a skillet over medium-low heat. Add the reserved caramelized onions, and heat until warmed. Spoon over dumplings, and serve immediately with sour cream and applesauce.