Using a food processor or blender, blend egg, lemon juice, mustard, garlic, salt, sugar, and cayenne for a few seconds.
While blender on low speed slowly add vegetable oil until mixture is thick, a couple minutes. Scrape down bowl sides with spatula and blend for a few more seconds.
Transfer aioli to mason jar for easy storage. Add olive oil and stir until incorporated. Cover and refrigerate until needed. At least an hour.
Enjoy every spoonful!
Notes
Aioli can be refrigerated covered in an airtight container for up to 1 week.