1. Preheat the oven to 425°. Toast the bread directly on the oven racks until dry and lightly golden, about 3 minutes. Rub 1 side of the toast with the cut sides of the garlic clove. Lower the oven temperature to 375°.
2. Brush the bottom of a 9-by-13-inch baking dish with 1 tablespoon of the reserved artichoke oil and arrange one-third of the toast in a single layer. Top with half of the artichokes. Season lightly with salt and pepper and top with one-third of the cheese. Repeat with another layer of toast, artichokes and cheese and season with salt and pepper. Top with the remaining toast and cheese.
3. In a bowl, mix the milk with the eggs. Season with salt and pepper. Pour the custard over the bread; cover with plastic wrap. Lay a few cans on the plastic to keep the bread submerged. Let soak until most of the custard is absorbed, about 15 minutes. Remove the plastic.
4. Place a sheet of oiled parchment paper on top of the pudding and cover with foil. Bake in the center of the oven for 30 minutes. Remove the foil and parchment; bake for 15 minutes longer, or until the top is golden. Let the pudding cool for 15 minutes before cutting into squares and serving.
Make Ahead: The pudding can be prepared through Step 3 and refrigerated overnight. Let stand at room temperature for 30 minutes before proceeding.
Wine Recommendation: Look for a full-flavored Chardonnay with only a little oak, such as the 2002 Tormaresca from Italy or the 2000 Antonin Rodet Château de Chamirey Blanc from France.