Wash artichokes by plunging them up and down in cold water. Cut off stem ends, and trim about 1/2 inch from top of each artichoke. Remove any loose bottom leaves. With scissors, trim sharp edges from the top of outer leaves, if necessary.
Place artichokes in a large Dutch oven; cover with water, and add 1/2 cup olive oil and next 4 ingredients. Bring to a boil. Cover, reduce heat, and simmer 40 minutes or until lower leaves pull out easily.
Combine remaining 1/2 cup olive oil, balsamic vinegar, and next 2 ingredients in a blender. Process until well blended. Set vinaigrette aside.
Combine mayonnaise, pesto, and 3 tablespoons lemon juice. Set aside.
Drain artichokes, and place on lettuce leaves. Drizzle each with 2 to 3 tablespoons of vinaigrette and dollop with pesto sauce. Garnish, if desired.