In the top of a double boiler or a heat-safe bowl set over gently simmering water, vigorously whisk together the egg yolks, 1 tablespoon of water, and the lemon juice. Continue whisking steadily until the mixture thickens slightly and reaches a safe temperature of 160°F—this should take about 30 to 40 seconds.

Immediately remove the bowl from the heat and place it into a larger bowl filled with ice water. Keep whisking continuously for another 1 to 2 minutes until the mixture cools down completely.

Pour the cooled egg mixture into a narrow container, such as a 2-cup glass measuring cup. Add the salt and a dash of white pepper. Using an immersion blender, begin blending the mixture while slowly streaming in the olive oil. Blend steadily until the mayonnaise becomes thick, creamy, and fully emulsified.

If a lighter consistency is desired, whisk in the remaining 1 tablespoon of water. Transfer the finished mayonnaise to a sealed container and refrigerate. Use within 7 days.