Bring water to a boil in a wide pot and salt well.
Quickly mix all the ingredients for the dough with a food processor and a dough hook. The dough does not have to be completely smooth.
Press the dough into the boiling water with a spaetzle slicer or a spaetzle press and stir to prevent gnocchi from sticking together. When the gnocchi float to the top, use a strainer to scoop them out of the cooking water into a bowl of cold water, then drain in a colander.
Lightly brown butter in a frying pan. Toss 2/3 of the sage leaves and the hazelnuts in it. Add the gnocchi and season with sea salt and pepper.
Place on a plate or in a bowl and sprinkle with the remaining sage leaves and nuts.