Peel mirlitons; cut in half lengthwise, and discard pits. Cut mirliton halves lengthwise into slices. Steam mirliton slices and artichoke bottoms, covered, 40 minutes or until tender. Cool.
Combine vinegar and next 7 ingredients (vinegar through pimento) in a large bowl; stir with a whisk.
Add vegetables to vinegar mixture; toss well. Cover and marinate 1 hour, stirring occasionally. Spoon into a lettuce-lined bowl, if desired. Serve with a slotted spoon.