This is how to make aioli the traditional way.

Mince and mash garlic to a paste with a pinch of salt using a large heavy knife. Set aside.

Whisk together yolk, lemon juice, vinegar and mustard in a bowl. Combine oils and add a few drops at a time, to yolk mixture, whisking constantly, until all oil is incorporated and mixture is emulsified. (If mixture separates, stop adding oil and continue whisking until mixture comes together, then resume adding oil.)

Whisk in garlic paste, vanilla a few drops at a time to taste, and herbs, then season with salt and pepper. If aïoli is too thick, whisk in 1 or 2 drops of water. Chill, covered, until ready to use.

Here is how to make it using a food processor or blender.

Mince garlic through a press or with a heavy knife. Add egg yolk, hard-boiled egg yolk, chopped into pieces, lemon juice, vinegar, mustard and 1 tablespoon of olive oil. Pulse until mixed well. With the motor running, add the oil in a very slow stream until the oil is incorporated and the mixture is emulsified. This can happen fairly quickly. Stop and taste the mixture. Add salt and pepper to taste and adjust oil/acid ratio if necessary. Add vanilla, a few drops at a time to taste, then herbs.

If the sauce breaks, scrape it out of the processor bowl, wash the bowl, add an egg yolk, and then add the broken sauce very slowly until it emulsifies then add garlic, vanilla and herbs. This is mostly likely to happen during humid weather or a thunderstorm.

Store as you would mayonnaise. This will keep for about a week.