In a food processor, pulse the egg, lemon juice, mustard, salt and and pepper until combined.
With the machine running, very, very slowly drip in the oil until the mayo thickens.
Taste and add additional salt if desired.
With an immersion blender
Place all ingredients minus oil in a tall container (the container the blender came with is ideal)
Using a hand immersion blender, blend until smooth
While blender is on, very, very slowly drip in the oil and blend until the mayo thickens.