For the egg gnocchi, bring water to a boil in a large pot. As soon as the water boils, add salt. In the meantime, melt the butter. Quickly mix the melted butter with the egg, flour, salt and water to form a smooth dough.
Using a small spoon, quickly prick dumplings from the dough and slide them into the boiling water. Do not take too much time, because the dough will quickly become tough. The dumplings are ready as soon as they float to the surface of the water.
Drain the water, put the gnocchi in a pan and pour some melted butter over them. Beat the two eggs, season with salt and pepper and pour over the hot dumplings. While doing this, stir the egg gnocchi well until the egg mixture is set. Sprinkle with chives and serve.