Steaming: This can be done in advance to save time. Steamed artichokes can be stored in the refrigerator for up to 1 week. To trim for steaming: Cut stem of artichoke. Cut 1/3 of top off artichoke off, discard. Clip off tips of each leaf with scissors. Stand artichokes in steamer basket with 3 inches of water, cover and boil until fork tender at base, about 25 to 40 minutes depending on size.
Marinating: Split cooked artichokes lengthwise, scoop out "choke" (fuzzy center). Mix oil, balsamic, garlic, rosemary, lemon juice and salt and pepper, in the bottom of glass baking dish. Place artichokes, cut side down, in marinade. Cover and refrigerate until ready to grill, up to a day.
Grilling: Remove artichokes from marinade, save marinade for basting, place artichokes on hot grill cut side up and cook until deep golden brown, about 5 to 7 minutes. Brush with marinade and turn to cook other side until deep golden brown, brush again with marinade. Serve hot or at room temperature.