Place the potatoes in a large pot and cover fully with cold water.
Add the caraway seeds and a large pinch of salt to the water and place over medium heat.
Once the water comes to a boil, reduce heat to a simmer and cook until the potatoes are completely tender, 20-30 minutes.
Remove from heat, transfer the potatoes to a colander, and set aside until just cool enough to handle.
Peel the potatoes and press through a potato ricer into a large bowl.
Fold in the potato starch, salt, and nutmeg just until evenly distributed.
Mix in the egg yolks, egg, and butter just until a soft and smooth mixture comes together.
Form the dough into individual balls about 2 1/2 inches (4 centimeters) wide and set aside, repeating with the remaining dough.
Bring a large pot of lightly salted water to a boil.
Reduce heat to a low simmer (it should not be boiling at all) and add a few of the Kartoffelknödel, taking care to not overcrowd the pot.
Very lightly stir along the bottom to keep the Knödel from sticking and simmer until they have risen to the top and are heated completely through the center, about 15-20 minutes.
Carefully remove with a slotted spoon and repeat with remaining Kartoffelknödel.
Serve immediately.