Add the egg, mustard, lemon juice, salt, and pepper to bowl or tall jar and pour oil on top and use immersion blender, keeping blender flat for 20-30 seconds.
Start slowly lifting until all oil is incorporated and mayo is fluffy.
Taste mayo for seasoning then add salt, lemon juice, or extra vinegar to taste for extra bite if desired.
Or use a food processor:
Use the small bowl of processor or a blender for emulsion to work. Mix all ingredients minus oil for 20 seconds. Slowly drizzle, adding oil a bit at a time and blending until all oil is added.
Once your mayo is done, set aside covered and refrigerate.
Boil Eggs:
Carefully place eggs in a large pot of cold salted water, add a splash of vinegar to help prevent cracking, and a teaspoon of baking soda to help with peeling
Bring to a slow boil on medium heat. Remove from heat and cover for 12-15 minutes.
Get a large bowl of iced water ready to receive the hard-boiled eggs.
With a large spoon carefully add eggs to ice water to cool them off and help with peeling.
Fill Eggs:
Carefully slice eggs in half lengthwise and pop out the yolks and add to a large bowl. Set whites in a tray to stay upright.
Add the mayo, mustard, vinegar, salt, and pepper to the egg yolks and mash together with a fork until well mixed.
Add salt, lemon juice, or extra vinegar to taste for extra flavour.
Spoon each egg half with mixture and garnish with paprika, chives, or dill weed.
Serve immediately or refrigerate leftovers for up to 3 days.
Notes
You will only need half of the mayo made, so store unused mayo in fridge for up to 2 weeks in an airtight jar.
You can stir in crumbled bacon or use as a garnish.
Chopped sweet pickles, green olives, or smoked salmon are also great options.