1. Wash artichokes by plunging up and down in cold water. Cut off stem ends, and trim about 1 inch from top of each artichoke. Remove and discard any loose bottom leaves. Trim and discard one-fourth off top of each outer leaf with scissors.
2. Bring artichokes and water to cover to a boil in a Dutch oven; cover, reduce heat, and simmer 25 minutes. Drain; pat dry with paper towels.
3. Cut artichokes in half lengthwise. Remove choke using a small spoon or melon baller. Snap off and discard tough ends of asparagus.
4. Stir together olive oil and next 3 ingredients. Brush cut sides of artichoke halves with one-third of olive oil mixture. Brush asparagus evenly with one-third of olive oil mixture, reserving remaining olive oil mixture for later use.
5. Grill artichokes and asparagus, covered with grill lid, over medium-high heat (350° to 400°). Grill artichokes, cut sides down, 10 minutes; turn and grill 5 more minutes. Grill asparagus 1 to 2 minutes; turn and grill 2 more minutes or until tender. Garnish, if desired. Drizzle with reserved olive oil mixture, if desired, and serve immediately.