In a small pot of boiling salted water, cook the potatoes until fork-tender, about 8 minutes. Drain and transfer to a medium bowl. Mash with 2 tablespoons butter; let cool. Stir in the cheddar cheese and season with salt and pepper.

In a large bowl, combine the flour, 1 teaspoon salt and 1/2 teaspoon pepper. Make a well in the center of the flour, add the egg and sour cream and stir into the flour. Let the mixture rest for 10 minutes, then transfer to a work surface and knead until a smooth, stiff dough forms. Cut the dough into quarters, wrap each in plastic and refrigerate for 30 minutes.

Meanwhile, in a large skillet, cook the remaining 1 stick butter over medium heat until melted and brown, about 5 minutes. Stir in the chives.

On a floured work surface, working with one-quarter of the dough at a time, roll out each piece into an 11-inch circle. Using a 4-inch biscuit cutter, cut the dough into rounds. Place a tablespoon of potato filling slightly off center on each circle. Moisten the edge, fold the dough over the filling and press the edges firmly to seal.

In a large pot of boiling, salted water, cook the pierogi, about 9 at a time, for 5 to 7 minutes. Toss them in the brown butter before serving.