Preheat a grill to medium-high heat.
Trim the stem from each Artichoke to 1-inch long, then bend back and snap off dark outer leaves. Cut top inch off of Artichokes with serrated knife. Using a vegetable peeler, peel dark green areas from stem and base of Artichoke. Halve each Artichoke. Using a small, sharp knife, cut out the choke and remove the purple, prickly tipped leaves from the center of each wedge. (Or you can wait until after the Artichoke is steamed to remove the choke. It'll be softer and easier to remove, but hotter since it's right out of the pot.) As you finish prepping your Artichokes, rub a slice of Lemon along all cut and exposed parts to help prevent browning (although I've found that slight browning is inevitable.)
Once finished, place into pot and add 1-2" water to the bottom of the pot. Steam Artichokes on medium-high heat for 12-15 minutes.
Remove the steamed Artichokes and place onto preheated grill (or if you waited to remove the choke, remove then place on grill.) Cook until tender and lightly charred in spots, turning occasionally, about 10 minutes.
Meanwhile, in a medium sized bowl, add Aioli ingredients and mix well.
Remove Artichokes from the grill and drizzle with Olive Oil, sprinkle with Salt, and serve with Lemon Garlic Aioli.