Add the egg to a small food processor bowl and process for 20 seconds. Add the salt and vinegar and process for 20 seconds more.
After cleaning the bowl's sides and bottom, switch on the food processor, and start adding the oil carefully, a little at a time, until about a quarter of it has been added (this is critical for proper emulsification).
You can relax your rules a bit after you see that the mixture is starting to thicken and emulsify. Continue adding it gradually while the processor is running, but increase the amount to a very thin stream rather than drips of oil.
After adding all the oil, scrape the bowl's bottom and sides, then process for an additional 10 seconds. Once the mayonnaise has been properly seasoned, add salt, lemon juice, or additional vinegar as desired.
Remember to gradually feed in extra oil while the processor is running if the mayo seems too thin.
NOTES
Homemade mayonnaise can be kept for up to a week in the refrigerator when covered. Don't freeze it, as it will separate when thawed.
Choose fresh, well-refrigerated, clean grade A or AA eggs with undamaged shells when picking raw eggs for homemade mayonnaise. Use pasteurized eggs if raw eggs are a concern.
Olive oil: Olive oil can be a touch overbearing, so choose one that is light and fruity. You might also use olive oil for only half of the oil and something more neutral for the other half.
Immersion blender method: Put the egg, salt, vinegar and oil in a tall, narrow jar that just fits the blender head. Start blending at the bottom in short pulses, then slowly move the blender up and down until thick.
Fixing broken mayonnaise: If it splits, put one extra egg yolk in a clean bowl and slowly blend the broken mayo into it until it thickens again.
Note that this recipe calls for no mustard and no added water. Just follow the steps and you'll get the thick, creamy mayonnaise you want.