Break egg into bottom of 1-quart canning jar or other tall narrow jar. The jar should be only slightly wider than the end of the stick blender.
Add lemon juice, vinegar, mustard, table salt and white pepper.
Add 1 cup of vegetable oil.
Place mixing blades of stick blender (turned off) all the way to the bottom of the jar, pressing down over the egg.
Turn stick blender on high speed, hold in place at bottom of jar for about 5-seconds, until you see mayonnaise form under stick blender's mixing blades.
Slowly pull stick blender upward until the mixing blades reaches top of jar, taking about 5-seconds to do so. The stick blender will turn the oil into mayonnaise as it is pulled slowly to the top of the jar
Chill before using to allow mayonnaise to thicken slightly and for flavors to develop.
I use this pretty quickly and always remember it does not have preservatives in it. Being as it makes one cups has never been a concern. Have never tried using egg substitute for it should work ok if you want to try.