Add the egg, vinegar, dried mustard, salt, and avocado oil to a wide-mouth mason jar or the cup that came with your immersion blender.
Press the immersion blender head firmly against the bottom of the jar so it fully covers the egg.
Keeping the blender at the bottom, blend until a thick, white, creamy mayo forms at the base — about 30 seconds.
Once that thick layer forms, slowly lift and lower the blender to incorporate the remaining oil.
Continue blending until everything is smooth, thick, and fully emulsified — another 1 to 2 minutes.
Store in an airtight container in the refrigerator for up to 1 week.