Halve cooked artichokes lengthwise; scoop out center petals and fuzzy centers. Remove outer leaves and reserve to garnish salad; or refrigerate to use for snacks another time. Thinly slice hearts. Cover; set aside.
Place whole bell peppers under preheated broiler; broil under high heat until charred on all sides, turning frequently with tongs. Carefully remove from oven with tongs; place in a paper bag for 15 minutes to steam skins. Carefully remove peppers from bag; remove seeds and ribs. Strip off skins; slice peppers into julienne strips.
Prepare Dressing: In blender or food processor container, place 1/4 of the bell pepper strips, vinegars, garlic, basil, rosemary. Cover and process until well blended and nearly smooth.
Assemble Salads: Arrange reserved cooked artichoke leaves or lettuce leaves on 8 salad plates. Arrange sliced artichoke hearts, remaining 3/4 of the bell pepper strips, red onion and olive slices on artichoke or lettuce leaves.