To make béarnaise sauce, you use the same technique as for hollandaise sauce, replacing the lemon juice with a reduction of wine, vinegar, shallots, and tarragon.
To make the reduction: Combine the shallots, wine, vinegar, peppercorns, and tarragon in a heavy-based saucepan and simmer over medium high until 2 Tbs. of liquid remains. Strain and discard the solids.
To make the sauce: In a heavy-based saucepan, melt the butter. Simmer it rapidly for at least 10 minutes.; the water will evaporate and the milk solids will coagulate on the bottom and sides of the pan. Let the melted butter sit for a few minutes so the solids will fall to the bottom. Skim off the foam on top and then either decant the golden liquid, leaving the solids behind, or pour the melted butter through a cheesecloth-lined strainer. Put the eggs and water in a Windsor pan (a saucepan with sloped sides) or in a heavy-duty metal mixing bowl (you'll have to hold it by one edge with a kitchen towel over the burner). Off the heat, whisk the eggs and water for 30 seconds, whipping in lots of air. Cook the sabayon over very low heat, whisking constantly and scraping the bowl, until thick and voluminous. The whisk will leave tracks that hold for a few seconds. At this point, take it off the heat and whisk rapidly for 30 seconds to cool it slightly. Add the clarified butter a little at a time, whisking constantly. Be sure the butter isn't too hot or it will break the emulsion.
To finish the sauce is done, whisk in 1 Tbs. of the reduction (or more to taste). Season with salt and pepper. Stir in finely chopped tarragon, if you like.
Photo: Scott Phillips