Melt the butter in a non-stick skillet over medium heat. Add the bread cubes and fry until golden on all sides. Move them to a bowl and set them aside for later.
Place the whole, unpeeled potatoes in a large pot of lightly salted water. Boil until they are tender all the way through. Keeping the skins on stops the potatoes from soaking up extra water.
Drain the potatoes and let them sit until you can handle them. They should still be warm. Peel off the skins, which slip off easily at this stage.
Press the warm potatoes through a potato ricer or food mill into a large bowl. This step is the secret to a light texture. Mashing them instead will give you dense, heavy dumplings.
Add the potato starch, salt, egg, and nutmeg to the riced potatoes. Knead gently with your hands until you get a soft, smooth dough. Don't overmix, and add a little more starch if the dough feels sticky.
Divide the dough into twelve balls, each about 2 1/2 inches wide. I like to weigh them, since each should be around 90 grams. Even sizes mean even cooking.
Press your thumb deep into the center of each ball to make a small well. Tuck two or three croutons inside. Pinch the dough closed and roll it back into a smooth ball.
Bring lightly salted water to a rapid boil, then turn the heat down. Lower each dumpling in on a spoon and hold it for a few seconds before letting go. Simmer uncovered for a full 20 minutes.
Turn off the heat once the timer ends. The dumplings can rest in the hot water for up to 15 minutes while you finish dinner. Move them to a warm plate and sprinkle with parsley.