Use a ceramic, glass, or stainless steel bowl (aluminum and copper will react with the acid and affect the color and flavor of the sauce). For easier hand-whisking, place the bowl on a damp kitchen towel to prevent it from rocking or spinning as you work.

Have ready a balloon whisk and a medium bowl or affix the whisk attachment to a stand mixer. Whisk together until smooth and light:

If using a stand mixer, increase the speed to medium-high. Very gradually whisk in:

As the mixture thickens—when about one-third of the oil has been added—whisk in the oil more steadily, making sure each addition is thoroughly blended before adding the next. Should the oil stop being absorbed, stop pouring the oil and whisk vigorously before adding more. Whisk in:

Serve immediately, or refrigerate for up to 1 week.

Place all the ingredients from version I in a narrow jar just wide enough to fit the end of an immersion blender (most immersion blenders come with such a jar—use it if possible). Place the end of the blender into the jar so it touches the bottom. Start the blender on medium speed, keeping it at the bottom of the jar until you can see the mixture form a vortex, thicken, and become opaque. Slowly lift the blender up and pulse it a couple times up and down to make sure all the oil is incorporated.