We will start by finely chopping the shallots. We put them in a small saucepan and add the vinegar and white wine plus the black peppercorns. We boil until reduce to a third of its volume. Remove from the heat, strain, remaining with the liquid and let it cool.
While doing this process we can clarify the butter common, putting it in another saucepan with the heat to the minimum. The solids will precipitate to the bottom and on the surface we will have the clarified and clean butter.
In a bowl placed in a bain-marie, beat the yolks with the mixture of vinegar, wine, shallot and pepper that we have strained. With the rods, we are giving zig zag movements until emulsified. Little by little, we are adding the clarified butter, almost by thread, without stopping to stir.
Almost without realizing it the mixture will begin to take shape, like mayonnaise while we continue adding the butter until finished. We also add the chopped tarragon and we taste and correct the salt and we have our béarnaise sauce ready to use.