Mince the leaves of one of the sprigs of tarragon. Set aside and discard the stem.

In small saucepan, combine the white wine, white wine vinegar, 3 full sprigs of the tarragon and shallot. Bring to a low simmer for about 15 minutes, remove from the heat.

Place the eggs, salt and white pepper in tall, but small cup or bowl that is just a tad wider that the head of an immersion blender. Blend.

Place a fine mesh sieve over the cup with the egg mixture and pour the vinegar mixture through, pressing out liquids from solids. Discard solids. Blend the egg yolks and vinegar mixture well.

While blending, slowly pour the melted butter into the egg mixture.

Simmer water in the bottom pot of a double boiler, place the bearnaise sauce in the top bowl and whisk constantly until it thickens, approximately 3-4 minutes. Remove from heat immediately and fold in the reserved minced tarragon.

Serve hot.

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