With a whisk or mixer running at low speed in the combined ingredients, gradually add the oil in a very slow, steady stream. If you add the oil slowly enough, an emulsion will form. (Your mayonnaise will thicken up.)
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The key to success with mayonnaise is to add your oil in a very slow, steady stream. As your mayonnaise begins to thicken up, you can add the oil a little faster. You must be patient, though, and pour slowly at first.
If your mayonnaise separates and does not thicken, you can rescue it! In a clean container with a clean whisk, mix ½ tablespoon cold water with another egg yolk. Add your (failed?) mayonnaise/oil mixture very, very slowly as you run the mixer or blender at low speed. It should then thicken up for you.