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.
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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.
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Remove from the heat, add the lemon juice, chopped tarragon, salt, and pepper and whisk to blend. Adjust the seasoning, to taste.
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Serve immediately or keep warm, covered, over a pot of simmering water, for 5 to 10 minutes. Whisk before serving.
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Yield: about 1 1/2 cups