Place the egg, lemon juice and mustard in a high-speed blender or food processor (or if you want to use an immersion blender, in a large mason jar).
Turn on the blender/food processor and very slowly drizzle in the oil until the mixture begins to emulsify and looks like mayonnaise. Continue adding the oil. Adjust seasonings.
Let sit at room temperature (not too hot!) for about an hour before using or storing in the refrigerator. Will keep about two weeks.