Prepare the Mixture: In a tall, narrow bowl or jug, whisk together egg, egg yolks, vinegar, lemon juice, salt, sugar, mustard, and cayenne.
Emulsify with Oil: Place the immersion blender into the bowl, ensuring the blades touch the egg mixture. Pour the avocado oil into the bowl. Hold the blender down and blend at high speed. Wait for the emulsion to occur (this should happen quickly). Once emulsified, slowly lift the blender to incorporate more oil.
Incorporate Remaining Oil: Continue blending until the mayonnaise is thick and well-emulsified, with some oil still unmixed. Pull the immersion blender up and down to incorporate the remaining oil.
Season to Taste: Adjust seasoning according to your taste preferences.
