Step 1At least 3 hours and up to the night before you plan to grill, drain the oil from the artichokes into a small bowl. To prepare the marinade, stir the juice of 1 lemon and 3 of the garlic cloves into the oil. Place the chicken in a resealable plastic bag and pour the marinade over it, rubbing some of the liquid under the skin without tearing the skin. Refrigerate the chicken.
Step 2Remove the chicken from the refrigerator and drain it, discarding the marinade. Let the chicken sit uncovered at room temperature for 20 to 30 minutes. Blot any excess liquid from the surface.
Step 3Fire up the grill, bringing the temperature to medium.
Step 4While the grill heats, puree the artichoke hearts in a blender with the olive oil. Warm the butter in a small nonreactive saucepan over medium heat and add the remaining garlic. Cook the garlic briefly until just soft and then stir in the wine, oregano and pureed artichoke-and-oil mixture. Cook the sauce for several minutes, bringing it to a good simmer. Stir in the remaining lemon juice, taste, and add salt to taste. If the sauce seems too thick, add a splash of wine and simmer an additional minute. Remove the sauce from the heat, but keep it warm. (The sauce can be made a day ahead, kept refrigerated, then reheated.)
Step 5Transfer the chicken to the grill skin-side down. Grill the chicken uncovered over medium heat for 30 to 35 minutes, turning 4 times. Watch for flare-ups, shifting the breasts away from the flame if necessary. If grilling covered, cook for 24 to 28 minutes, turning twice, though the time may vary depending on the size of the chicken breasts. End the grilling in either case with the chicken skin-side down to give it a final crisping. The chicken breasts are done when opaque down to the bone and still juicy.
Step 6Stir the sauce to reblend and spoon it onto a platter. Top the sauce with the chicken breasts, and scatter the basil or parsley and the lemon zest over the chicken.