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Make the infused vinegar. Add the wine, vinegar, shallot, and tarragon to a small saucepan over medium-high heat. Bring to a boil, then reduce the heat to a simmer, until the liquid has reduced to about 1½ tablespoons (about 5 to 7 minutes).
Strain it through a fine-mesh sieve, using the back of a spoon to extract as much flavor as possible into the liquid. Let it cool for 5 minutes.
Melt the butter in a microwave (make sure sure the cup or bowl is covered as it will splatter) for about 1 minute, until it's hot. Alternatively, you could heat it on the stove.
In a high-powered blender, add the egg yolks, infused vinegar, lemon juice, and salt. Blend for 5 seconds until combined.
Stream in the butter. With the blender running on medium-high, slowly stream the hot butter through the top lid cap opening and into the mixture until it's emulsified.
Add fresh herbs. Pour the bearnaise sauce into a small bowl, stir in the finely chopped tarragon, and serve while warm.