Butter a 9-by-13-inch glass baking dish. Set aside.

Bring a large stockpot of salted water to a boil over high heat. Rub the cut sides of the artichokes with the lemon halves. Squeeze the juice from the lemon halves into the water and add the lemon shells. Add the artichoke halves, cover and cook until just tender, 15 to 20 minutes. Drain in a colander, stem sides facing upwards, to allow any excess water to drain out from the leaves.

Heat the oil in a large skillet over medium-high heat. Add the garlic and cook until golden brown, 2 to 3 minutes. Remove the garlic and discard. Remove the pan from the heat and let the oil cool slightly. Whisk in the anchovy paste until smooth. Return the pan to the heat and place the artichokes, cut-side down, in the pan. Cook for 6 minutes. Turn the artichokes over and cook until tender, 6 minutes longer. Remove the artichokes and reserve the pan juices.

Preheat a broiler. Arrange the artichokes, cut-side up, in the prepared baking dish. Sprinkle with the salt, pepper, olives and capers. Mix together the cheese and breadcrumbs in a small bowl and sprinkle on top. Spoon the reserved pan juices over the breadcrumbs. Broil until golden brown, 2 to 3 minutes.