Combine the shallots, tarragon sprigs, white wine, and vermouth in a small saucepan, bring to a boil, and cook until reduced to 4 teaspoons. Strain through a fine-mesh sieve and let cool.
In the top of a double boiler, or in a metal bowl fitted over a pot of simmering water, whisk the egg yolks and reduced wine mixture until ribbons start to form when the whisk is lifted. Whisking constantly, drizzle in the melted butter, a bit at a time, until the mixture thickens.
Remove from the heat, add the lemon juice, chopped tarragon, salt, and pepper and whisk to blend. Adjust the seasoning, to taste.
Serve immediately or keep warm, covered, over a pot of simmering water, for 5 to 10 minutes. Whisk before serving.
Yield: about 1 1/2 cups