In a small saucepan, combine dry white wine, white wine vinegar, sliced shallot, and tarragon. Bring the mixture to a boil over medium heat, then reduce the heat to low and simmer for about 5-7 minutes.

Once the mixture has reduced, take it off the heat and strain it using a fine-mesh sieve into a bowl. Allow the strained liquid to cool for about 5 minutes.

In a microwave-safe bowl, add the unsalted butter and cover it. Microwave on high for about 1 minute or until very hot.

Add the cooled infused vinegar, egg yolks, lemon juice, and kosher salt to a high-powered blender. Blend the mixture on high for about 5 seconds until well combined.

With the blender running on low, slowly drizzle in the hot melted butter until the sauce thickens and becomes glossy—about 30 seconds.

Transfer the béarnaise sauce to a serving bowl, and gently stir in the additional chopped tarragon. Serve warm.