Put the flour, melted butter, eggs and a pinch of salt in a kettle and mix well with the milk. Work into a dough that is not too firm, adding a little more milk if needed.
In a large pot, bring enough salted water to a boil.
Immediately drop the dough (otherwise the mixture will gelatinize) through a spaetzle slicer directly into the boiling water.
As soon as the dumplings float to the surface of the water, strain and rinse with cold water. Stir in a few drops of oil to keep the gnocchi from sticking together.
Heat butter or clarified butter in a pan and add gnocchi, tossing constantly.
Season eggs with salt, pepper and a little nutmeg and mix briefly. Pour over the dumplings and let them brown for no longer than 2 minutes.
Arrange egg gnocchi on warmed plates and garnish with chopped chives.