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1
Beat the eggs until light and aerated. Add the flour, salt, and milk or water. Mix until just combined — the batter should fall in thick ribbons from a spoon but must not be worked further. Over-mixing develops gluten and produces tough, elastic noodles. Rest the batter for 10 minutes uncovered; it is ready when it flows from the spoon in a slow, unbroken ribbon.
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2
For Käsespätzle, melt 40 g butter in a wide pan over the lowest possible heat. Add the sliced onions and cook, stirring occasionally, for 45 minutes until they have collapsed to a deep amber, jammy mass. The sweetness of properly caramelised onions is the defining flavour of the dish against the sharp cheese and rich noodles; they are done when every strand is dark gold and the pan smells of caramel rather than raw allium.
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3
Bring a large pot of water to a full rolling boil. Salt it aggressively — Spätzle cooked in unsalted water taste of nothing regardless of subsequent seasoning. Working in small batches, press the batter through a colander or Spätzlehobel held directly over the water; the resistance of the holes breaking the soft batter produces the characteristic irregular, ear-like pieces. Do not overload the pot: a crowded pot drops temperature and the noodles cook unevenly and stick.
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4
This is where the dish lives or dies. Watch the surface of the water: each Spätzle is done the moment it bobs to the top. Remove it immediately with a slotted spoon — even thirty seconds beyond floating produces a gummy, overcooked centre. Transfer each batch directly into a warmed bowl and toss at once with a knob of butter; without fat, the noodles cool and fuse into a solid mass.
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5
For Käsespätzle: while the noodles are still hot, layer them in a warmed gratin dish with the grated cheese, finishing with a cheese layer on top. The residual heat of the noodles must melt the cheese into the pasta — cold noodles produce uneven, clumpy distribution. Spoon the caramelised onions over the surface and serve at once.