Beat eggs, milk, and seasonings in a medium bowl. Stir in flour to form a smooth but thick batter; let batter rest for 10 minutes.
Put 1 tablespoon butter into a bowl that has been rinsed in hot water and dried. Heat water to boil in a kettle or saucepan small enough so that the short ends of the spaetzle machine can rest on its rim.
Salt boiling water, then spoon a portion of the batter into the the square container that runs along the grater track. With the machine resting on the pan rim, move the metal container quickly back and forth along the grater until about 1/6 of the dough is pressed through the grater into the boiling water.
With a slotted spoon transfer spaetzle that have floated to the water's surface to the warm bowl. Repeat cooking in batches with remaining batter, adding butter to each batch of cooked spaeztle. Toss and serve.