Boil the potato until cooked, and cool under cold water.
Peel and crush the garlic and put into a blender along with the two egg yolks, and the juice from half a lemon. Add the cool potato and blend for a few seconds.
Put the blender on its lowest speed and initially very slowly pour in the olive oil until the sauce is very thick and yellow (may not use all the oil).
Season the aioli with salt and a little more lemon juice to taste.