Williams-Sononma.com and BonAppetit.com each have recipes for aioli that require less than 20 minutes to prepare. The Williams-Sonoma recipe calls for Dijon mustard, which helps to create an emulsification while imparting a bit of tang.
For the Williams-Sonoma recipe, whisk together 1 tablespoon of olive oil, 1 teaspoon of Dijon mustard and one egg yolk until the ingredients emulsify. Combine 1/3 cup each of olive oil and safflower or peanut oil in a pitcher, bottle or small wine decanter. Add the 2 to 3 tablespoons of the oil mixture drop by drop to the egg emulsification while whisking constantly.
When the mixture thickens, add the remaining oil in a fine, steady stream. Whisk constantly, incorporating all the oil. Stir in the juice of half of a lemon, 3 finely minced cloves of garlic and salt and pepper to taste. Thin the aioli with 2 tablespoons of warm water to use it as a sauce.
To make the Bon Appetit recipe, whisk together 1 egg yolk, 1 finely grated clove of garlic, 1/4 teaspoon of salt and 2 tablespoons of water. Drizzle in 1/4 cup of grapeseed oil one teaspoon at a time while whisking constantly, until the sauce emulsifies and thickens.
While whisking, drizzle in 1/4 cup of olive oil one teaspoon at a time. Stir in a pinch of cayenne, and add fresh lemon juice, salt and pepper to taste. A well made aioli should resemble runny mayonnaise, with aioli sauce being slightly thinner.