Melt 1 Tbsp. unsalted butter in a small saucepan over medium heat. Add 3 medium shallots, finely chopped, and stir to coat; season with kosher salt and freshly ground black pepper. Stir in 2 Tbsp. Champagne vinegar or white wine vinegar, reduce heat to medium-low, and cook, stirring occasionally, until vinegar is evaporated, about 3 minutes. Reduce heat to low and continue to cook, stirring often, until shallots are softened and translucent, about 5 minutes longer. Scrape shallot reduction into a small bowl and let cool.

Meanwhile, fill a blender with hot water to warm it. Melt remaining 1 cup (2 sticks) unsalted butter, cut into ½" pieces, in a clean small saucepan over medium heat. Once butter starts to foam, pour into a measuring glass.

Pour off water from blender; dry well. Add 2 large egg yolks, 1 Tbsp. fresh lemon juice, and 1 Tbsp. water and blend until mixture is smooth. With motor running, very gradually stream in melted butter, leaving milk solids at bottom of measuring glass behind. Blend until a smooth, creamy sauce forms.

Pour sauce into a medium bowl. Stir in shallot reduction and 1 Tbsp. finely chopped tarragon. Taste and season with more salt, pepper, and/or lemon juice if needed. Do Ahead: Sauce can be made 1 hour ahead. Cover and let sit at room temperature. Note: Raw egg is not recommended for infants, the elderly, pregnant individuals, or those with weakened immune systems.