To prepare artichokes: Squeeze juice from lemon half into a large bowl of cold water. Trim the very bottom of the stem. Snap off several of the bottom 3 rows of leaves until you reach the more tender leaves. Cut off artichoke tip with a chef’s knife to remove the pointy leaf tips (approximately 1 inch). Halve artichoke lengthwise. Using a spoon or melon baller, scrape out the small, inner choke. Place artichoke half in lemon water. Repeat with remaining artichokes.

Bring a large pot of salted water to a boil. Add artichoke halves and boil until tender, about 5 to 6 minutes. Drain in a colander and pat dry with paper towels. In a large mixing bowl, whisk together vinegar, oil, and garlic. Add the warm artichoke halves, tossing well to coat with marinade. Set aside for at least 30 minutes.

Prepare grill for medium heat or heat a grill pan over medium heat. Remove artichokes from marinade, remove excess liquid and reserve the marinade. Season with salt and pepper and place on grill cut side down. Cook until distinct grill marks develop, turning once, about 7 to 8 minutes total. 

To serve: Place grilled artichokes on a serving platter, drizzle with the reserved marinade and lemon aioli.  Serve immediately.