Boil potatoes in their skins. Drain and set aside until cool enough to handle. Scrape skin from potatoes with a paring knife. Press potatoes through a ricer, and spread into a thin, even layer on a clean work surface. 

In a small bowl, beat together the egg, salt, pepper, and nutmeg. Gather cooled potatoes into a mound and form a well in the center. Pour egg mixture into the potato well. Knead the potato and egg mixture together with both hands, gradually adding the grated Parmesan and about 1 ½ cups of flour. Mix with your hands until mixture has formed a smooth, slightly sticky dough. Add more flour as needed. 

Place the dough on a lightly floured surface, and cut it into 6 equal pieces. Form dough into a rope, then slice rope into 1-2–inch thick rounds, resembling fingers. Sprinkle the rounds with flour, then press each round lightly with the tines of a fork. Set dumplings on a baking sheet lined with a floured kitchen towel. Repeat with remaining dough. 

Fill a large saucepan with salted water, and bring to a boil. Drop dumplings into water, about 6 at a time. Cook, stirring gently, about 1 minute after they rise to the surface of the water. Remove dumplings with a slotted spoon and set aside. 

In a small skillet, heat the butter until melted. Add the bread crumbs, and cook until butter has browned, then remove from heat. 

Serve dumplings with browned butter and bread crumbs sprinkled on top.