Wash artichokes by plunging up and down in cold water. Cut off stem end, and trim 1/2 inch from top of each artichoke. Remove any loose bottom leaves. With scissors, trim away one-fourth of each outer leaf.
Combine breadcrumbs, cheese, garlic, and oregano; mix well. Spread leaves of each artichoke apart, and spoon breadcrumb stuffing onto leaves.
Place stuffed artichokes in a large Dutch oven with about 3 inches of water. Pour vegetable oil evenly over tops of artichokes. Cover and cook over medium heat 1 hour.
Using a slotted spoon, transfer artichokes to a serving platter, and garnish with lemon twists.