Combine the base.
In the container of your blender or food processor, add the room temperature egg, ground mustard, salt, and cayenne pepper.
Blend for about 30-60 seconds until combined and slightly frothy.
Begin the emulsion. With the machine running on low, begin adding the oil EXTREMELY slowly. Start with just a few drops at a time, then move to a very thin, steady stream. This is the most critical step for creating a stable emulsion.
Continue until you've added about half of the oil.
Add acid and finish.
Once the mixture has thickened noticeably, pour in the lemon juice and blend to combine. With the machine still running, continue to slowly stream in the remaining oil until the mayonnaise is thick, creamy, and glossy. Scrape down the sides as needed.
Store.
Transfer the mayonnaise to an airtight container or jar and store it in the refrigerator. It will keep for up to one week.
Drizzle the oil slowly. We can't stress this enough!
Adding the oil too quickly, especially at the beginning, will prevent the mixture from emulsifying. A slow, thin stream is the key to success.