Bring a large pot of salted water to a boil. Add the artichokes and cook over medium-low heat for 15 minutes. Remove from the heat and drain.
Preheat the oven to 325°F.
Add the tomatoes, bread, basil, and Parmigiano Reggiano to a medium bowl. Slowly add the olive oil, season with salt and pepper, and gently toss to combine.
Place the artichokes in a small baking dish. Gently stuff the artichokes with the mixture. Put the artichokes in the oven and cook for 1 hour. Remove from the oven.
Transfer the artichokes to a large platter or individual plates. Sprinkle additional Parmigiano Reggiano over the top. Serve immediately with the lemon wedges on the side.