In a small saucepan over medium heat, combine ½ cup of dry white wine, ¼ cup of white wine vinegar, 1 finely chopped shallot, and 2 tablespoons of tarragon leaves. Bring this mixture to a gentle boil, then reduce the heat and let it simmer for about 5-7 minutes until it reduces to 1½ tablespoons. Strain it into a bowl to remove solids and allow it to cool slightly.

Melt 1 cup of unsalted butter in a small saucepan over medium-low heat until it is fully liquid and hot, but not boiling. While the butter melts, add 3 egg yolks, the cooled infused vinegar, 1 tablespoon of lemon juice, and a pinch of kosher salt into a high-powered blender. Blend for about 5 seconds to combine all ingredients effectively.

With the blender running on low, slowly drizzle in the hot melted butter through the opening of the blender lid. Continue this process gradually over about 30 seconds, which will help create a thick and creamy emulsion for your Béarnaise sauce.

Remove the blender from its base and add in 1 tablespoon of finely chopped fresh tarragon. Use a spatula to stir gently, ensuring the tarragon is evenly distributed throughout the béarnaise sauce. Serve warm over grilled steak.