How to Make It
Squeeze the juice from 1 lemon half into a bowl of water. Working with 1 artichoke at a time, discard the dark green outer leaves. Cut off the top 1 inch of the artichoke, then peel and trim the bottom and stem. Halve the artichoke lengthwise and scoop out the furry choke. Rub with the remaining lemon half and add to the bowl of lemon water.
Preheat the oven to 400°. In a saucepan of simmering water, cook the artichoke halves until barely tender, 7 minutes. Drain well and pat dry.
Lightly oil a ceramic baking dish. In a skillet, heat 1/2 cup of oil. Add the garlic and cook over moderately high heat until golden, 4 minutes; discard the garlic. Add the anchovies and stir until dissolved. Add the artichokes cut side down and cook until crusty on the bottom. Flip them and cook until lightly browned, 3 minutes; transfer the artichokes cut side up to the baking dish. Let the garlic-anchovy oil cool slightly.
In a bowl, mix the bread crumbs and cheese. Stir in the garlic-anchovy oil and season very lightly with salt and pepper. In another bowl, mix the olives and capers with a pinch of pepper. Fill the artichoke halves with the olive mixture and top with the bread crumbs. Bake for 12 minutes, until the topping is browned and crisp and the artichokes are tender. Let stand for 5 minutes, then serve.