Begin by cracking the whole large egg into a tall immersion blender cylinder jar or cup, then mix. (Use a sufficiently tall cylinder for the immersion blender to prevent splattering during blending.)
Add white wine vinegar, Dijon mustard, salt, and a dash of black pepper or cayenne into the beaten egg. Blend for a few seconds to combine. Remove the immersion blender, allowing the egg mixture to settle for about a minute.
Once the egg mixture has settled, pour the oil into it. Let it sit briefly once more, allowing the oil to separate from the other components.
Once the oil has separated, swiftly insert the immersion blender entirely at the bottom of the cylinder jar or cup and do not move it.
Next, activate the immersion blender while keeping it fixed at the bottom of the cylinder. Refrain from moving it until the mixture transforms into a creamy mayonnaise consistency. Exercise patience as this transformation may take a few minutes.
As the mixture emulsifies and thickens, gently lift the immersion blender, thoroughly integrating the remaining liquids in the cylinder.