Prepare ingredients: Finely chop 3 tablespoons shallots. Measure 3 tablespoons dry white wine and 2½ tablespoons white wine vinegar. Cut 9 tablespoons cold unsalted butter into cubes for gradual incorporation.

Reduce liquids: In a medium heavy-bottomed saucepan over medium heat, melt 1 tablespoon butter. Add shallots and cook gently for 1 to 2 minutes until translucent but not browned.

Add wine and vinegar: Stir in white wine and white wine vinegar and cook, stirring occasionally, until reduced to about 2 to 3 tablespoons (au sec), concentrating the flavors.

Incorporate cream and salt: Add ⅓ cup heavy cream and 1/8 teaspoon kosher salt. Bring to a boil, stirring, and cook for 1 minute to slightly thicken the base.

Strain sauce: Pour the mixture through a fine mesh strainer into a small bowl, pressing to extract flavors. Rinse the saucepan to prepare for emulsification.

Emulsify butter: Return the strained liquid to low heat. Whisk vigorously while slowly adding the remaining 8 tablespoons cold butter, a few cubes at a time, until the sauce is smooth, creamy, and coats the back of a spoon resembling heavy cream.

Season and serve: Add more salt and a pinch of white pepper if desired. Serve the sauce warm immediately. It can be kept warm for up to one hour, whisking occasionally. For reheating, do so gently while whisking and add a splash of cream if needed to prevent breaking.