Trim the artichoke.  Artichokes have pokey tips on the leaves and it is much more pleasant for those handling them and eating them if you remove the sharp tips of the leaves.  Take a sharp chef’s knife and cut the top quarter inch off the artichoke, then cut the very tips off each leaf.  Trim the stem.  Rinse the artichoke thoroughly under cold water.

Fill a large pot with about 2 inches of water along with 1 tablespoon salt and the juice of 1/2 lemon. Bring to a boil. Add the artichokes; cover and steam until the artichoke bottoms pierce easily with a fork, about 40 minutes. Drain the artichokes and set aside on a cooling rack.  Let cool.

When cool enough to handle, cut the artichokes in half, lengthwise, and scrape out the fuzzy center.  Make ahead tip:  At this point you could easily cover the artichokes and refrigerate them until you are nearly ready to serve.

Prepare a grill.  In a small bowl, stir together the juice of 1/2 lemon, garlic, and olive oil. Season with salt and freshly ground black pepper. Brush the artichokes with the garlic-oil mixture and set cut-side down, on a grill over medium heat. Grill, turning once, until lightly browned, about 6-8 minutes total.

You can serve these artichokes with melted butter, mayonnaise or you can dress up mayonnaise for a little added flavor by making a dipping sauce.  To make the dipping sauce, place the shallots, capers, mayonnaise and lemon juice in a small bowl and stir to combine.  Season to taste with salt and freshly ground black pepper.  Serve sauce alongside grilled artichokes.