The method is not my own invention but I have used it for years and it works.
You need a deep but narrow container and a hand blender. The container needs to be just wide enough for the blender to fit in and sit flush on the base.
Add the egg (room temperature), salt, vinegar and mustard to the container. Do not stir or break the egg yolk. Now add 400ml oil. Allow this to rest until the egg and other solids are sitting on the bottom.
Gently place the hand blender into the container. With it touching the bottom, press the button and do not move it. Eventually you will start to see mayo forming like a lava lamp up the side of the container. Now you can start to gently move the blender upwards.
Now you are at this point, its ok to be a bit rougher. Mayo will be ready in about 30 seconds and should last about 10 days in the fridge unless you follow the next few steps.
At this point, if I have some whey to hand I add about a tablespoon and stir it in. I then decant into a jar and leave it in a warm place for a good 7 hours. The mayo should taste tangy when it goes into the fridge. Fermented whey will last much longer – up to 6 weeks in the fridge and contains all those good bacteria the adverts tell us about!
You can google the method and find some great videos on how to do it.