Beat the egg whites until foamy. Combine the cheese with the egg yolks and flour in a separate bowl. Fold in the beaten egg whites to form a dough. Cover and refrigerate until nice and firm, at least 2 hours.
Generously dust a clean work surface with flour. Working in batches, take a large spoonful of the dough and roll into a log about 1 inch in diameter. Gently pat the log to flatten it. With a sharp knife, cut the log on a bias about every 1 inch into diamond-shaped pierogies. Repeat with the remaining dough.
Bring a pot of lightly salted water to a boil. Working in batches, slip the pierogies into the water and boil until they float to the surface, about 2 minutes. Do not overcook them. Gently transfer the pierogies to a plate using a slotted spoon or spider.
While the pierogies are cooking, melt the butter in a small nonstick skillet over medium heat. Add the breadcrumbs, stirring occasionally, and toast until golden.
To serve, spoon some of the toasted breadcrumbs, then a dollop of sour cream on top of each pierogi. Sprinkle with sugar. Smacznego! Bon appetite!