In a large mixing bowl, combine flour and salt. Create a well in the center and add the egg and melted butter. Gradually add warm water while mixing, starting with 3/4 cup and adding more as needed until a soft, pliable dough forms.

Knead the dough on a lightly floured surface for 8-10 minutes until smooth and elastic. The dough should be soft but not sticky. Wrap in plastic wrap and let rest for 30 minutes at room temperature.

Boil the quartered potatoes in salted water until fork-tender, about 15-20 minutes. Drain thoroughly and mash until completely smooth. Let cool to room temperature.

Meanwhile, heat 1 tablespoon butter in a skillet over medium heat. Add diced onion and cook, stirring frequently, until golden brown and caramelized, about 8-10 minutes. Let cool.

Combine the cooled mashed potatoes, well-drained cottage cheese, caramelized onions, salt, and white pepper. Mix thoroughly and taste for seasoning, adjusting as needed.

Divide the rested dough into 4 portions. On a floured surface, roll each portion into a thin circle, about 1/8-inch thick. Using a 3-inch round cutter or glass, cut circles from the dough.

Place 1 tablespoon of filling in the center of each circle. Fold the dough over to create a half-moon shape, then press edges firmly with a fork or crimp with your fingers to seal completely. Ensure no air pockets remain.

Place finished pierogi on a floured baking sheet, making sure they don't touch each other.

Bring a large pot of salted water to boil. Gently add pierogi in batches, stirring immediately to prevent sticking. Cook until they float to the surface, about 3-4 minutes, then cook 2 minutes more.

Remove with a slotted spoon and drain briefly. For extra flavor, heat remaining 2 tablespoons butter in a large skillet over medium-high heat. Pan-fry the boiled pierogi until golden brown on both sides, about 2-3 minutes per side.

In the same skillet, add sliced onion with 2 tablespoons butter. Cook over medium heat, stirring occasionally, until deep golden brown and caramelized, about 15-20 minutes.

Serve hot pierogi topped with caramelized onions, a dollop of sour cream, and fresh chives.