To make the aioli: Beat the yolk with the vinegar and mustard in a bowl. Whisk in the oil, drop by drop, to make a thick mayonnaise. Squeeze the garlic out of the roasted cloves into the mayonnaise and whisk smooth. Season the aioli with salt, pepper, and lemon juice, to taste.
To prepare the artichokes: Trim the artichokes, rubbing with lemon as you go, until you are down to the core with a bit of stem attached. Halve. While you're trimming, bring a pot of water to the boil. Squeeze the juice of a lemon into it and salt it. Drop in the artichokes and cook until tender, about 10 minutes, depending on the size. Drain, and pat dry. Heat the olive oil in a skillet and brown the artichokes on all sides. Transfer to paper towel to drain. Serve with the aioli.
Cooks Note* To roast a head of garlic, wrap in foil and bake for an hour at 350 degrees F.