Trim the artichokes:  Remove the outer layer of small, tough leaves from the stem end.   Trim the stem to create a flat bottom and discard cut stem.  With a very sharp serrated knife (like a bread knife), cut off the top 1 inch of each artichoke.  Use scissors to snip the tips of the remaining leaves.  Cut the artichoke in half through the stem end.

Cook the artichokes:  put a steamer insert in a large pot of water and bring water to a simmer.  Add artichoke halves to steamer insert and cover.  Steam artichokes until stem can be pierced easily with a knife, about 25-35 minutes depending on size.  Remove artichokes from the saucepan and set aside until cool enough to handle.  Preheat the grill on medium heat.

Make the sauce:  In a medium bowl combine all the sauce ingredients and taste for seasoning.

Remove the chokes:  Pull out the lighter leaves in the center of the artichoke to expose the fuzzy choke.  With a spoon, scoop out the choke (but not the heart) and discard.

Brush inside and outside of the artichoke with olive oil, sprinkle with salt and pepper and grill until char marks appear, about 5-10 minutes.  Flip over and grill on other side.  Serve with lemon-caper dipping sauce.