Aioli, sometimes called the "butter of Provence" because it is so integral to the cuisine, is simply a mayonnaise incorporating quantities of fresh garlic - usually two cloves per person in Provence. That is a lot, and you might wish to use less (I do - three or four cloves for this recipe), but since the whole point of aioli is garlic, by all means don't stint. The quality of the garlic is most important - it should be fresh and plump with swollen cloves. Reject any shriveled cloves or any in which the core is developing a green sprout.
Traditionally aioli is made with egg yolks and garlic pounded in a mortar. A lighter sauce can be made in the blender with a whole egg and an egg white, but the blender does odd things to the taste of garlic, so I prefer to make the lighter mayonnaise in the blender, then fold the garlic paste in separately by hand. I always use organic eggs, but if you're not sure of your egg source and worried about salmonella, don't try this recipe.
1 whole egg
1 egg white (you can freeze the yolk and use it later)
salt to taste
1 to 1 1/2 cups fruity extra-virgin olive oil as needed
juice of 1/2 lemon or more to taste
4 garlic cloves, coarsely chopped
Make the mayonnaise by whirling the egg and egg white in a blender with a little pinch of salt. Remove the center knob from the blender lid and, with the blender churning, start to pour in the olive oil, a very thin thread at first, until the mixture starts to thicken. Stop the blender and pour in a few tablespoons of lemon juice, then start the blender again and continue adding oil, a little more thickly as the mixture emulsifies and mounts. When all the oil has been added, turn the blender off and set aside.
Combine the garlic in a small bowl or a mortar with a teaspoon of salt and pound the garlic or crush it with the back of a spoon until you have a thick and homogeneous paste. Now use a spatula to scrape the mayonnaise into the garlic paste and turn gently to incorporate everything. Taste and add more salt and lemon juice if desired.
If the mayonnaise breaks down and separates while you're blending it, remove it all from the blender and start over again with a fresh egg. While the egg until it is light and add, bit by bit, the broken-down mayonnaise and more oil and lemon juice. It should be easy to reconstitute the mayonnaise. Then procede as directed