Place the cubed potatoes in a large pot of salted water. Bring it to a boil. Cook until the potatoes are very soft. This usually takes about 15 minutes. Drain the water well.

Mash the hot potatoes until they are smooth. Add the shredded cheese and butter while they are warm. Stir until the cheese melts completely. Add salt and pepper. Let the mixture cool down.

In a large bowl, whisk the egg and sour cream together. Add the flour, salt, and softened butter. Stir with a wooden spoon until a shaggy dough forms. Use your hands to bring it together.

Turn the dough onto a floured surface. Knead it gently for about 5 minutes. The dough should feel soft and elastic. Do not overwork it. Let it rest for 30 minutes under a towel.

Divide the dough into two pieces. Use a rolling pin to roll it out thin. It should be about 1/8 inch thick. Make sure the surface stays floured so the dough does not stick.

Use a 3-inch round cutter to cut circles from the dough. You can also use the rim of a glass. Keep the scraps to re-roll later. You should have plenty of circles.

Place a small spoonful of the potato filling in the center of each circle. Do not overfill them. If you add too much, they might burst during cooking. Keep the edges clean.

Fold the dough over the filling to make a half-moon shape. Pinch the edges firmly with your fingers. You can also use a fork to press the edges. This creates a pretty pattern.

Bring a large pot of salted water to a boil. Drop 6 to 8 pierogi into the water. They will sink at first. Wait until they float to the surface. Cook for 2 more minutes.

Remove the pierogi with a slotted spoon. Heat a skillet with butter over medium heat. Fry the pierogi until they are golden brown on both sides. Serve them hot with extra sour cream.