Combine the ingredients for the aioli in a small mixing bowl, adding salt and pepper to taste. Cover and refrigerate while you prepare and grill the artichokes.
Rub the cut sides of the artichoke halves with a half a lemon. Add the lemon to a pot of boiling water. Parboil the artichokes until tender, about 20 minutes. Drain, then submerge in cold water. Drain again, then arrange cut sides down on paper towels.
Combine the cumin, black pepper, and sea salt in a small bowl. Brush the artichokes on all sides with olive oil and season with the rub. Set up the Kalamazoo Hybrid Fire Grill for wood grilling according to the manufacturer’s instructions. Grill the artichokes directly over the embers, cut sides down, for 2 to 3 minutes, or until nicely browned. Turn, and grill the leaf side for 2 to 3 minutes.
Serve the artichoke halves, warm or cold, with the aioli.
