For those of you who witnessed our spirited discussion, Jeff and I have come to a draw on the aioli. We've figured out that the egg can be optional-- olive oil + garlic, mashed with a pestle-- but the breadcrumbs are as well. Why add breadcrumbs to a sauce that has egg in it already? Bread's a binder! Breadcrumbs must go into the olive oil and garlic version. Mustard is added to some versions, but not to others. There is obviously no definitive answer. Further research and testing is required. Oh, darn.
(The further I research this, the more muddled it gets. I just found a recipe that involves both breadcrumbs and egg yolks. I think the only real conclusion is that aioli is not mayonnaise. Ta da.)
Some pictures from the evening, courtesy 5chw4r7z: