Mayo will emulsify more easily if all ingredients are at room temperature or slightly warmer. If your egg is refrigerated, place the egg (still in its shell) in a coffee cup and fill it with hot tap water. Set aside for five minutes while you gather the remaining ingredients.
Place the egg, apple cider vinegar, ground mustard, and salt in a wide mouth pint size mason jar. Add MCT oil (if using) up to the 4 ounce marking on the jar. Then add avocado oil up to the 10 ounce marking.
Place the immersion blender in the jar. Make sure the egg yolk is under the blender, the blender is touching the bottom of the jar, and the blender is parallel with the sides of the jar.
Turn the blender on high. Leave it in place until there begins to be emulsification at the bottom of the jar, approximately 10 seconds.
Tilt the running blender until it is resting on the side of the jar.
Slowly pull the running blender up through the jar. This should take about 30 seconds. Move very slowly to allow the blender to gradually pull the oil into the emulsion. If you go too fast, your emulsion will break.
After all the oil is pulled in, turn off the blender and remove it from the jar. There might be small pockets of unemulsified oil. I like to stir the mayo a bit with a mini whisk to make sure it is smooth and creamy throughout.
Top the mason jar with its lid and band and store the mayo in the fridge for up to a week.
Rescuing Broken Mayo
If your mayo is only slightly broken, like it looks almost emulsified, but not quite, or if it was emulsified at first but then broke as the blender neared the top of the jar, you may be able to rescue it with your mini whisk. Simply whisk the broken mayo until it becomes smooth and creamy.
If whisking didn't work, add another egg to the broken mayo and repeat steps 3-8 above. Make sure to move VERY slowly when pulling the blender up through the mixture. The resultant mayo will be slightly thinner than usual, but it will still work in recipes.
Notes
If you are concerned about consuming raw eggs, you can buy pasteurized eggs or pasteurize them at home.