Part 1: Preparing the Artichokes

1. Prepare an Ice Water Bath: Fill a large bowl with ice water and squeeze the juice of half a lemon into it. This will prevent the artichokes from browning after they are cut.
2. Trim the Artichokes: Working with one artichoke at a time, snap off the tough outer leaves around the base until you reach the pale green and tender inner leaves. Cut off the top inch of the artichoke crown. Use kitchen shears or a vegetable peeler to trim the sharp tips off the remaining outer leaves. Trim the stem flush with the base, or leave about an inch of stem if you prefer a handle for grilling and eating. Immediately rub all cut surfaces of the artichoke with the cut side of the remaining lemon half to prevent oxidation.
3. Halve the Artichokes: Stand the trimmed artichoke upright on its base. Using a sharp knife, cut the artichoke lengthwise in half, right through the stem.
4. Remove the Choke: Using a small spoon or a melon baller, scoop out the fuzzy choke (the hairy part in the center) and any prickly inner leaves from each artichoke half. Be thorough in removing all of the choke as it is not edible. Rub the cut surfaces again with lemon juice.

Part 2: Boiling the Artichokes

1. Boil Water: Fill a large pot with water, enough to submerge the artichokes. Add the smashed garlic cloves, salt, and peppercorns to the water. Bring to a rolling boil over high heat.
2. Boil Artichokes: Gently place the prepared artichoke halves into the boiling water. Reduce the heat to medium-low, cover the pot, and simmer for 15-20 minutes, or until the artichoke hearts are tender when pierced with a fork. The cooking time will depend on the size and freshness of your artichokes.
3. Drain and Cool: Carefully remove the artichokes from the boiling water using tongs or a slotted spoon. Plunge them into the ice water bath to stop the cooking process and cool them down quickly. Let them cool in the ice bath for a few minutes, then drain them well and pat them dry with paper towels.

Part 3: Grilling the Artichokes

1. Prepare the Grill: Preheat your grill to medium-high heat. If using a gas grill, preheat with all burners on medium-high, then reduce to medium heat for grilling. If using a charcoal grill, prepare a medium-hot fire. Clean and oil the grill grates.
2. Brush with Olive Oil: In a small bowl, combine the olive oil, salt, and pepper. Brush the cut sides of the boiled artichoke halves generously with the olive oil mixture. Also brush the halved lemon you reserved with olive oil.
3. Grill Artichokes: Place the artichoke halves cut-side down on the hot grill grates. Place the lemon halves cut-side down on the grill as well. Grill for 4-6 minutes per side, or until the artichokes are nicely charred and grill marks appear. Grill the lemon halves for about 3-4 minutes, or until slightly softened and grill marks appear.
4. Remove from Grill: Remove the grilled artichokes and lemon halves from the grill and set aside.

Part 4: Making the Lemon Garlic Aioli

1. Combine Aioli Ingredients: In a small bowl, whisk together the mayonnaise, minced garlic, fresh lemon juice, lemon zest, Dijon mustard, salt, and pepper. Whisk until smooth and well combined.
2. Adjust Seasoning: Taste the aioli and adjust seasoning as needed. Add more lemon juice for tanginess, salt and pepper to taste, or a pinch of cayenne for heat if desired. You can also stir in chopped fresh herbs or capers at this point if using.
3. Chill (Optional): For the best flavor, cover the aioli and refrigerate for at least 30 minutes to allow the flavors to meld. Aioli can be made up to a day in advance and stored in the refrigerator.

Part 5: Serving the Grilled Artichokes

1. Serve Warm with Aioli: Arrange the grilled artichoke halves on a platter. Serve immediately while warm, accompanied by the lemon garlic aioli for dipping.
2. Garnish (Optional): Garnish the platter with the grilled lemon halves, fresh parsley sprigs, or extra lemon wedges. Encourage guests to squeeze the grilled lemon juice over their artichokes for an extra burst of smoky citrus flavor.