If you’re using an immersion blender, place the mustard, salt and vinegar in a small bowl or mason jar with a wide enough mouth to fit the blender inside. If you’re using a food processor or blender, place the ingredients in the associated appliance bowl or container. (If pasteurizing your egg, I find it easiest to complete this step before the pasteurization process so that it doesn't slow me down once my egg is ready).
If you did the pasteurization step, carefully remove the egg from the hot water and crack the egg into the bowl with the other ingredients. You may see that some of the white has started to cook through – that’s okay, it will blend together. If you did not do the pasteurization step, simply crack the egg into the bowl with your other ingredients.
Turn the blender on and mix the ingredients until well combined.
While still blending, pour a tiny amount of oil into the bowl and blend thoroughly (see note if your appliance does not allow you to add ingredients while blending).
Repeat this process, pouring a tiny amount and mixing thoroughly until you see the mixture start to change color and the bubbles start to go away. Check out the photo (or watch the video) for how it should look at this stage.
Begin pouring the oil in a steady stream while you hold the blender with the other hand. The mixture will emulsify and change to a pale creamy white color, and it will start to develop the texture of mayonnaise that you’re familiar with.
Continue until all the oil has been poured into the bowl and blend until the texture is thick. For me, I know that the mayo is perfect when I can lift the immersion blender up slowly and back down and it creates the shapes shown in the photo.
Transfer the mayo to a tightly sealed container (or simply place the lid on the mason jar and refrigerate. The mayonnaise will thicken and set after a few hours in the fridge. However, you can also use it immediately, it will just be warmer and a looser texture.