Cut the stems off from the bottom of the artichoke, but leave about 1 inch. Remove the small tough, loose leaves from the base and stem. Cut one inch off the top of the artichoke using a sharp paring knife.
Use kitchen scissors to trim the sharp points at the tips of the leaves. Repeat for the remaining artichokes.
Add salt to a large pot of water and bring to a boil. Add the artichokes, cover and simmer until they are fork tender at the base.
Slice the cooked artichokes in half. Use a small spoon to scoop out the dense, fuzzy choke at the center (right above the artichoke heart) . Remove all of the hairs and the small leaves.
Place artichokes in a bowl or large Ziploc bag. Combine the olive oil, balsamic vinegar, minced garlic, minced onion, salt, pepper, Italian seasoning, lemon juice, and the zest of 1 lemon to make the marinade, in a small bowl.
Pour the olive oil mixture over the artichokes. Cover or seal and refrigerate for at least 4 hours or overnight. Stir occasionally.
Heat the outdoor grill to high heat. Place the cut side of the artichoke on the hot grill grates in a single layer. Place the flat side of the lemon halves down on the grill and sear until lightly charred.
Squeeze the grilled lemon juice into a small bowl and whisk the mayonnaise in. Season with black pepper.
Garnish the grilled artichokes with fresh basil or a sprinkle of sea salt. Serve with the dipping sauce as a delicious appetizer or as an easy side dish at your next dinner party.