In a large bowl, combine mashed potatoes, shredded cheddar, sour cream, and egg. Mix until well combined.

Add flour, salt, and melted butter to the mixture. Knead until a smooth dough forms.

Roll out the dough on a floured surface to about 1/8-inch thickness.

Use a 3-inch round cutter to cut out circles from the dough.

Place a teaspoon of filling onto each circle, fold in half, and seal the edges by pressing with a fork.

Bring a large pot of salted water to a boil. Cook the pierogies in batches until they float, about 3-4 minutes.

Remove with a slotted spoon and serve hot with melted butter or your favorite toppings.