Step 1Place the minced garlic in a heavy mortar along with the salt. Pound with a pestle into a smooth, sticky paste. Using the pestle to stir, beat in the egg yolks and stir until the mixture is smooth and lemon colored, about 30 seconds.
Step 2Begin to beat in the olive oil, adding just a few drops at a time, stirring constantly until the oil is emulsified with the egg yolks. Do not attempt to rush the process by adding the oil too fast. After you've added about 1/4 cup this way, you can increase the flow to a thin stream. If at any time you see oil begin to gather separately from the yolks, immediately stop adding oil while continuing to stir. Very shortly the mixture should come back together.
Step 3When you have added about 1/2 cup of oil and the aioli is quite thick, stir in the water, then begin adding the oil again, starting again with a drop at a time.
Step 4When you've worked in about 3/4 cup oil, stop and taste the sauce. The texture should be creamy, not stiff and sticky. If it is too firm, stir in a little more water up to 1 teaspoon total. The flavor should be very garlicky but sweet, with a balance of garlic and olive oil flavor. If it's still a little too garlicky, stir in a little more oil. Add more salt if necessary and the lemon juice, if that is to your taste.
Step 5If the sauce breaks and the oil and yolks separate and won't come back together, it can be easily fixed. Add a whole egg to a blender and purée it until smooth. Pour the broken aioli mixture into a measuring cup and, with the blender running, slowly add it to the blender. When it is completely incorporated, slowly add more oil with the blender running until you have the texture and flavor you prefer.
Step 6Cover tightly and refrigerate until ready to use, but allow it to return to room temperature before serving. Aioli should be made no more than a couple of hours in advance.