Bring the water and salt to a boil in a 2-quart saucepan, add the potatoes and cook uncovered.
Select a heatproof plate the same diameter as the pot to serve as a lid. Lay the herring fillets on the plate side by side and make neat, diagonal slices through them every 1/2 inch without disturbing their original shape. When the potatoes are about half-cooked (after 1 minutes or so, but test now and then with the tip of a small, sharp knife), cover the pot with the plate of sliced herring and continue cooking the potatoes over medium heat for another 1 to 12 minutes or until potatoes are tender and the herring is heated through or 'melted'.
Meanwhile, melt 2 tablespoons of butter in a small saucepan. Cook slowly over moderately low heat to brown the butter lightly.
Drain potatoes, return to pan and dry over low heat for a minute or two, shaking constantly. Place the herring on a serving plate and garnish with chopped parsley, radishs and egg whites and yolks. Pour the hot butter on the herring only and serve the potatoes in a separate bowl.
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