Combine the shallots, vinegar, and tarragon in a saucepan and reduce by three-fourths. Remove from heat and cool slightly. Strain, if you wish.
Combine the vinegar mixture with egg yolks in a medium-sized, stainless steel bowl. Using a wire whisk, beat well. Place the container containing the egg/yolk acid mixture in a pan or bowl of water (such as a small double boiler) that is at a simmering temperature of 180 to 190F (82.2 to 87.8C). Continue whisking the yolk/acid mixture and heat to a temperature of 150F (65.6C). The mixture must be stirred or whisked constantly and the temperature measured frequently by an accurate thermometer. Immediately remove the pan containing the yolk/acid mixture from the hot-water heat source. The mixture should be creamy. Cool slightly.
Slowly and gradually beat the clarified butter into the yolk acid. Add the butter drop by drop at first to disperse the butter oil and create an emulsion. As the emulsion forms, the butter can be added in larger amounts. (If the sauce becomes too thick to beat before all the butter is added, beat in a little warm water.) Adjust seasoning with salt and cayenne pepper.
If necessary, thin the sauce with a few drops of warm water. Mix in the parsley and tarragon. The final pH is 4.2.
Storage Keep warm [100 to 120F (43.3 to 48.9C)], not hot, for service. This Béarnaise sauce can be held at 100 to 120F (43.3 to 48.9C) as long as the quality allows. There is no food safety problem because of the pH.
*Note: To clarify butter for this recipe, place 1 1/4 pounds butter in heavy sauce pan. Melt butter over moderate heat. Skim froth from the surface. Carefully pour off clear melted butter into another container, leaving the milky liquid and salt (if salted butter was used) on the bottom of the pan.
