Clarify the butter. (Melt and strain off the butter solids.) You should have about 1 pint clarified butter. Keep the butter warm but not hot.

Combine peppercorns, salt, and vinegar in a saucepan and reduce until nearly dry (au sec). Remove from heat and add the cold water. Pass the diluted reduction through a fine strainer into a stainless-steel bowl. Use a clean rubber spatula to make sure you transfer all flavoring material to bowl.

Add the egg yolks to bowl and beat well. Hold the bowl over a hot-water bath and continue to beat the yolks until thickened and creamy.

Remove the bowl from the heat. Using a ladle, slowly and gradually beat in warm clarified butter, drop by drop at first. If the sauce becomes too thick to beat before all the butter is added, beat in a little lemon juice.

When all the butter is added, beat in lemon juice to taste and adjust seasoning with salt and cayenne. If necessary, thin the sauce with a few drops of warm water.

Combine the shallots, vinegar, tarragon, and peppercorns in a saucepan and reduce by three-fourths. Remove from heat and cool slightly.

To make it easier to beat with a wire whip, transfer the reduction to a stainless-steel bowl. Use a clean rubber spatula to make sure you get it all. Let the reduction cool a little.

Add the egg yolks to the bowl and beat well.

Hold the bowl over a hot-water bath and continue to beat the yolks until thickened and creamy.

Remove the bowl from the heat. Using a ladle, slowly and gradually beat in the warm clarified butter, drop by drop at first. If the sauce becomes too thick to beat before all the butter is added, beat in a little lemon juice or warm water.