Add the garlic and salt to the mortar bowl and grind slowly with the pestle, moving in one direction only. You can do this first step in a food processor if you'd like, then transfer the mixture back to a medium sized bowl. Whisk in the mustard first, then the egg yolks. At this point you can transfer the mixture back into the mortar or use the whisk in the bowl.
Now you will add in half of the oil. This must be done very slowly or the oil will not emulsify and your sauce will not thicken. Add the oil in a slow, fine stream while either whisking with a wire whisk or using your pestle. Once the first half of the oil is incorporated, then add the water and the lemon juice, whisking or stirring constantly with the pestle. Then slowly add the rest of the oil. The mixture will thicken as you continue to blend it. The mixture should be slightly thinner than commercial mayonnaise. If it becomes too thick you can add a bit more warm water, one teaspoon at a time.
Your Aioli Separated, Now What?
Add a single egg yolk (room temperature) to a bowl and and whisk the separated aioli into it. The additional yolk will reimulsify the sauce.
Rosemary - Add two teaspoons of minced fresh rosemary
Chipotle - Add one tablespoon of minced canned chipotles in adobo sauce. Read more about Chipotles (smoked jalapeno chiles).
Saffron - Add a tiny pinch of Saffron threads and a tablespoon of honey.