If using a large food processor, attach the smaller bowl to ensure proper emulsification.

Add egg to the small bowl of a food processor and process for 20 seconds. Add mustard, vinegar, and salt, then process for another 20 seconds.

Scrape the sides and bottom of the bowl, then begin to slowly add the oil in tiny drops until about a quarter of the oil has been added.

When the mixture begins to thicken, continue adding the oil slowly in a thin stream.

Once all the oil has been added, scrape the bowl and process for an additional 10 seconds. Taste and adjust seasoning with salt, lemon juice, or extra vinegar.

Add egg, mustard, vinegar, lemon juice (optional), and salt to a tall, narrow jar, then pour all the oil over the ingredients.

Insert the immersion blender wand and press it against the bottom of the jar. Turn on high without lifting the blender until a thick mixture forms at the bottom (15-30 seconds).

Slowly lift the blender up and down to incorporate the remaining oil until thick and creamy. Taste and adjust seasoning.