While versions of it are made around the rim of the western Mediterranean, from Spain all the way to Sicily, this garlic sauce is best known in its Provençale incarnation known as aïoli. It is a commonplace to refer to aioli in English as garlic mayonnaise, but it is really something rather different.
You begin with a mortar and pestle. Add several whole cloves of garlic and a pinch of salt. (Besides seasoning the sauce, the salt helps to soften the garlic.) Begin by crushing the garlic cloves against the side of the mortar with your pestle, then starting turn the pestle around in the mortar, always in a single direction, until the garlic has been reduced to a rough paste. This can be slow work, so be patient…
Then add an egg yolk, mixing it into the garlic as you turn your pestle, and, drop by drop, begin to add olive oil, as you continue to turn the pestle. Don't add more oil until the last bit has completely emulsified into the sauce. Again, be patient… after a while, you will be able to add more oil at a time. Continue thus, turning always in the same direction, until the sauce is thick enough that your pestle will stand up on its own, like this:
Now, if you like—and I do like—you can temper the sauce with a few drops of freshly squeezed lemon juice. Adjust for seasoning and you're done!
At this time of year, I like to serve aioli with steamed seasonal vegetables: asparagus, artichokes, green beans, together with baby carrots and new potatoes still in their jackets. If you are perfectionist, you should steam them separately so that each will be done to just the right degree, having lost their rawness but still brightly colored and offering some slightly resistance to the bite. Lazy cooks like me, however, add all the vegetables at once on a steamer placed over simmering water in a large covered sauté pan and let them all steam together until the potatoes are done. (You will know when they are done when you can prick them with a paring knife and then remove the knife without picking up the potato.)
I like to serve this dish 'family style, by placing a small bowl with the aioli in the middle of a platter, then arranging the steamed vegetables around the platter. Each diner can spear a vegetable of their choice, dip into the sauce and eat. It's a lovely, convivial way to share a meal. But you can also serve the sauce separately, to be passed around to each diner to serve themselves—perhaps a better plan if you're not on intimate terms with your dining companions.
NOTES: Purists will tell you that the original aioli was made only with garlic, salt and olive oil—no egg yolk. But all the contemporary recipes I have seen, including the one in the Larousse Gastronomique, call for the egg yolk which, of course, greatly helps the emulsification process. (Some versions call for adding a bit of boiled potato or soaked bread instead of egg yolk, but they are a distinct minority.) And while the same purists will tell you that aioli is definitely not a kind of mayonnaise, you can make a rather close ersatz version by adding a few cloves of garlic to mayonnaise made in a blender or food processor. It's a practical, but not very romantic, solution. And the original mortar and pestle method, once you get the hang of it, is really quite easy. It only requires a modicum of skill and, most of all, a sense of patience. They say that making aioli while you're nervous is a surefire way for it to 'curdle'.
The best of olive oil to use for this dish is, of course, from Provence, where the olive oil is fruity but lighter than most Italian olive oils, especially the ones from Puglia and Sicily. But I've made the sauce from fruity Italian olive oil that suited me just fine. The most important point, perhaps, is to make sure that the garlic is as fresh as it can be. As for measurements, the Larousse calls for 4 cloves of garlic, one egg yolk and about 2.5 dl (1 cup) of olive oil. But being a garlic fiend, I sometimes add a clove or two more, depending on the size of the cloves.
For a true Provençale treat, the classic Grand Aïoli is made with desalinated and poached salt cod, boiled snails (actually, a special kind of snail called bulots), hard-boiled eggs and boiled vegetables, typically green beans, new potatoes, baby carrots, cauliflower, fennel, chickpeas, zucchini, beets... It makes for a sumptuous but healthy spring or summertime meal. At the opposite end of the spectrum, aioli is perfectly delicious as a starter with a single vegetable; to my taste, asparagus and artichokes are especially lovely covered with an unctuous dressing of aioli.
Post-scriptum: Contrary to what you may have read recently, the word aïoli is not Italian for mayonnaise, which is, quite simply, maionese. The word comes from ail (French for garlic) and oli, the Provencale dialect word for oil (huile in standard French). Because the word sounds Italian to many non-Italian speakers--Provencale dialect has a lot in common with Italian-it seems to create some confusion.