Prepare the dough: Put flour in a large shallow bowl and make a well in the center.

Combine wet ingredients: Add water, egg, oil, and salt to the well and carefully beat together with a fork without incorporating flour.

Incorporate flour: Continue stirring with a wooden spoon, gradually incorporating flour until a soft dough forms.

Knead the dough: Transfer the dough to a lightly floured surface and knead, dusting with flour as needed to keep the dough from sticking, until smooth and elastic, about 8 minutes.

Rest the dough: Invert a bowl over the dough and let it stand at room temperature for 1 hour.

Prepare potatoes: Peel potatoes and cut into 1-inch pieces.

Cook potatoes: Cook potatoes in a large saucepan of boiling salted water until tender, about 8 minutes.

Make the filling: Drain potatoes, then transfer to a bowl along with cheese, salt, pepper, and nutmeg and mash with a potato masher or a handheld electric mixer at low speed until smooth.

Shape filling: When mashed potatoes are cool enough to handle, spoon out a rounded teaspoon and lightly roll into a ball between the palms of your hands.

Store filling: Transfer ball to a plate and keep covered with plastic wrap while making 47 more balls in the same manner.

Caramelize onions: Cook onion in butter in a 4- to 5-quart heavy saucepan over moderately low heat, stirring occasionally until golden brown, about 30 minutes.

Season onions: Remove from heat and season with salt and pepper.

Roll out dough: Halve the dough and roll out 1 half on a lightly floured surface with a lightly floured rolling pin into a 15-inch round (1/8 inch thick).

Cut out rounds: Cut out 24 rounds with a lightly floured cutter.

Fill pierogies: Holding 1 round in the palm of your hand, put 1 potato ball in the center of the round and close your hand to fold the round in half, enclosing the filling.

Seal pierogies: Pinch edges together to seal completely. If edges don't adhere, brush them lightly with water, then seal.

Store formed pierogies: Transfer pierogi to a lightly floured kitchen towel and cover with another towel.

Repeat filling: Form more pierogies in the same manner.

Boil water: Bring a 6- to 8-quart pot of salted water to a boil.

Cook pierogies: Add half of the pierogies, stirring once or twice to keep them from sticking together, and cook 5 minutes from the time the pierogies float to the surface.

Coat in onions: Transfer the cooked pierogies with a slotted spoon to the onion topping and toss gently to coat.

Cook remaining pierogies: Cook remaining pierogies in the same manner, transferring to the onions.

Reheat and serve: Reheat pierogies in onion topping over low heat, gently tossing to coat before serving.