The recipe below is for traditional aioli, which is made with uncooked egg yolks. If you prefer to avoid raw eggs, a delicious variation can be made with commercially prepared mayonnaise, see Aioli-Style Mayonnaise.
2 tablespoons fresh lemon juice, plus more to taste
1/2 cup vegetable oil
In a mortar, pound the garlic cloves with 1 teaspoon of sea salt until a thick paste forms. Scrape the mixture into a glass or stainless steel bowl, stir in the egg yolks and let stand for 5 minutes.
Set the bowl on a damp kitchen towel (to hold the bowl steady) and begin whisking in the olive oil a few drops at a time. Slowly add 1/2 cup of the olive oil, whisking constantly; as the aioli begins to thicken, you can add the oil in a thin stream. When the aioli is very thick, add 2 teaspoons of the lemon juice. Gradually whisk in another 1/4 cup of the olive oil, then 2 more teaspoons of the lemon juice. Whisk in the remaining 1/4 cup of olive oil, add the 2 remaining teaspoons of lemon juice, then whisk in the vegetable oil. When all of the vegetable oil has been incorporated, season the aioli with lemon juice and sea salt to taste.
The aioli can be tightly covered and refrigerated overnight. Serve chilled.