This sounds a *lot* like what I know as thomeyya, or domeyya, or dumeya (it's a real problem writing Arabic words in English). Anyway, the stuff is terrific - although the version I know is extremely garlicky, so it must be a slightly different version than yours. I wish I could tell you I have the answer, but my efforts - based on recipes from various cookbooks and internet searches - don't come close to reproducing the recipe I'm after. I'm ending up with a lot of odd salad dressings because I can't bear to throw the stuff out. Garlic, salt, lemon juice and oil figure in all the recipes, and the rest is either mayonnaise or yogurt to thicken it - or neither - but in any case there isn't enough mayonnaise to turn it into aioli. The basic steps, in case you want to experiment, seem to be to whirl the garlic, salt and lemon juice into a paste, then start blending in oil, then blend in the thickener if you're going to try one. I'm even wondering about egg whites. I will certainly post back if I succeed in reproducing the version I'm after. In the meantime, keep googling around for "garlic dip" or "dhumeyya" or...well, you have to be creative with the spelling...