
In a heavy bottomed pot heat a couple of inches of vegetable oil until it reaches 325-350 degrees on a thermometer.
While it is heating, set out the flour in a pie plate or shallow dish, and the breadcrumbs in another. Put the beaten egg or the buttermilk in a small bowl
Dredge the olives in the flour, and then into the beaten egg or buttermilk.
Toss them into the breadcrumbs and make sure they get evenly coated.
Fry the olives in the hot oil until they are golden, this will take about a minute and a half. Do this in batches so the oil doesn't cool down when you add the olives.
Drain the olives on a paper towel, and serve hot. Toothpicks help with the dipping.
To make the aioli whisk together all the ingredients.
The olives can be reheated by putting in a 350F oven for a few minutes. They can even be briefly microwaved.