For the egg dumplings, first make the dough: Beat the flour, the 3 eggs, salt and a little milk with a large wooden spoon to make a viscous dough, let it rest a little. If necessary, add a little more milk. Pour the dough in portions into a spaetzle slicer and thus plane the dumplings into boiling salted water. Cook for a few minutes. Then drain in a large sieve and rinse with cold water.
Heat olive oil in a large frying pan. Add the well drained gnocchi and heat. Beat the 5 eggs together and add salt. Pour the eggs over the dumplings and let them set, stirring often. Serve the egg gnocchi sprinkled with fresh chives.