Prepare the artichokes by trimming off the tops and stems. Slice each artichoke in half and use a spoon to remove the fuzzy choke from the center. Immediately rub with lemon wedges to prevent discoloration.
Fill a large pot with water and bring to a boil over medium-high heat. Add the salt, smashed garlic cloves, and lemon halves to the water, squeezing the lemons first. Add the artichokes and boil for 15-20 minutes until softened and fairly tender.
Remove the artichoke halves from the water and let them dry on a baking sheet. Meanwhile, combine olive oil and minced garlic in a bowl. Brush each artichoke on all sides with the garlic and olive oil, then season with salt and pepper.
Preheat grill to medium-high heat. Place artichokes on the grill cut-side up. Grill for 3-4 minutes, then flip and grill cut-side down for another 3-4 minutes until tender and charred. Serve with your favorite dipping sauce.