Basically a form of mayonnaise, intended to be served at room temperature or chilled, I make this condiment (or sauce base) with roasted garlic. It's creamy and garlicky and goes perfectly with both shellfish and roast beef.
In a 450-degree Fahrenheit (230-degrees Celsius) oven, roast the peeled cloves of garlic in canola oil. Wrap the garlic cloves and oil, tightly, in tin foil. To make the garlic cloves soft like butter, this will take between 30 and 40 minutes. To prevent scorching, add a half teaspoon of water to the tinfoil: this will keep it from browning, however, and the taste won't be quite the same. But it should protect the delicate garlic from burning.
In a blender, place the roasted garlic with the egg, lemon juice, salt, and white pepper. Starting at medium speed, blend for about a minute until the garlic has been fully incorporated into the blend.
Start with a few drops, rising to a drizzle, and add the olive oil. Keep blending for about 5 minutes until the aioli comes together into a thick, white sauce.