Combine half the bunch of tarragon, the shallots, peppercorns, vinegar and wine in a small saucepan and bring to a boil. Reduce to a simmer and simmer until it has reduced by half. Let the mixture cool and strain through a fine strainer.

Bring a medium saucepan of water to a boil over medium high heat. Once the water has boiled, reduce the heat to a minimum and allow the water to cool for about 5 minutes.

Combine the tarragon reduction, egg yolks, lemon juice and salt together in a stainless steel bowl and whisk vigorously until the mixture has almost doubled in volume. Place the bowl over the simmering water and continue to whisk, making sure you don’t scramble the eggs. Remove the bowl from the saucepan if you suspect it might be getting too hot.

Continue to stir vigorously, adding the butter as slowly as you can. Continue until all the butter has been incorporated into the egg yolks. The sauce should be thick and smooth. Check the seasonings and add the cayenne pepper and remaining leaves of tarragon, finely chopped. Hold the Bearnaise in the bowl covered with plastic wrap, over a barely simmering water bath.