In a large mixing bowl, whisk together flour and salt. Create a well in the center, add the beaten egg, sour cream, and softened butter. Gradually mix until a smooth dough forms. Knead for about 5 minutes, then let it rest for 30 to 60 minutes.

While the dough rests, combine warm mashed potatoes and shredded cheddar cheese in a bowl. Season with salt and pepper to taste, ensuring the cheese melts slightly.

Roll out the rested dough to about 1/8-inch thickness. Cut into circles, place a teaspoon of filling in each, and fold to seal.

Bring a large pot of salted water to a boil. Drop perogies into boiling water, boil until they float, and cook an additional 2 to 3 minutes. Remove with a slotted spoon.

In a skillet, melt chilled butter and sauté diced shallots until translucent. Add boiled perogies and fry until golden brown on both sides.

Serve hot, optionally with sour cream, chives, or crispy bacon bits.