In a clean, dry bowl, vigorously whisk together the egg yolk, lemon juice, and Dijon mustard until pale and frothy.
Begin adding the avocado oil drop by drop, whisking constantly, until the mixture begins to thicken and emulsify.
Once the emulsion is stable, pour the remaining oil in a slow, steady stream while continuing to whisk vigorously.
If desired, gently fold in the finely pureed half avocado for extra richness and color.
Season to taste with sea salt and white pepper. Transfer to an airtight container and chill in the refrigerator for at least 30 minutes before serving.
