In a small saucepan, combine 2 tablespoons of the chopped tarragon, the minced shallots, 1 tablespoon vinegar, and 1/4 cup of wine over medium heat.
Simmer until reduced by about half--the easiest way to tell is to tip the pan to see the amount in the corner of the saucepan. Remove from heat and cool slightly.
In a food processor, blend 3 egg yolks, a pinch of white pepper, and the tarragon reduction. With the processor running, slowly add the butter in a steady stream until all 8 tablespoons have been added.
The sauce consistency should be like a thin gravy. For thicker sauce, gently heat the sauce on low in a saucepan while stirring constantly until desired consistency is reached.
Remove from the processor and whisk-in remaining fresh tarragon. Season with salt and pepper, to taste.
Sauce can be made in advance and kept warm set in a hot water bath until ready to serve, or refrigerate and gently reheat before serving.*Raw egg is not recommended for infants, the elderly, and pregnant women. To minimize the risk of salmonella, use pasteurized eggs available in some supermarkets or easily pasteurize eggs at home. Alternatively, the sauce can be gently heated.