Bring a large pot of salted water to a boil. Fill large bowl with cold water. Stir together flour and salt. Whisk together eggs and milk, then whisk into flour until batter is smooth.
Working over boiling water, force one third of batter through a spaezlemaker or large holes of a food mill. As dumplings float to surface, transfer them to bowl of cold water with a slotted spoon. Make 2 more batches in same manner. Drain dumplings well and toss with oil.
Pulse dill, spinach, chives and 1 tablespoon butter in food processor until paste forms. Heat remaining tablespoon butter in large non-stick skillet over moderately high heat until foam subsides, then saute dumplings, stirring, until golden. Add herb butter and saute, stirring, until dumplings are coated and heated through. Season with salt and pepper. Serves 2