In a small saucepan, combine the tarragon, shallots, vinegar and wine over medium-high heat. Bring to a simmer and cook until reduced by half. Remove from heat and set aside.
Place a stainless steel bowl in a saucepan containing simmering water, or use a double boiler. Whisk the egg yolks until doubled in volume. Slowly add the melted butter, continue beating until sauce is thickened. Stir in reserved shallot reduction. Season with salt and pepper, set aside wrapped in a warm spot.
Made this for Mother's Day, 2004. Bearnaise continues to vex me.... This year, it was better with this recipe, but not what I'd call perfect. I'd stongly suggest that in the future, the tarragon/vinegar part could be done in advance, but the sauce should be made once meat is already on the plate...... I made it about an hour in advance this year, and it was like polenta by the time I wanted to serve it.... I heated it, ever so lightly over some hot water, and stirred which brought it to thick pudding texture, smooth, didn't break, but thick. When I put it over the asparagus, it didn't melt, just sat there..... Later, I heated some leftover in the microwave for 10 seconds, and that broke.....