In a tall, narrow container (like a mason jar), add mustard, vinegar, eggs, salt, and lemon juice.
Place the immersion blender at the bottom of the container and blend until smooth.
With the blender running, slowly pour in the avocado oil in a thin, steady stream.
As the mixture thickens and emulsifies, gradually raise the immersion blender up through the mixture until all the oil is incorporated.
Blend just until it reaches a creamy, thick consistency.
Store in a jar in the refrigerator for up to a week.