Remove beans from pods; discard pods. Cook beans and snap peas in boiling water 1 minute or until snap peas are crisp-tender. Remove beans and snap peas with a slotted spoon. Plunge beans and snap peas into ice water; drain. Remove tough outer skins from beans; discard skins.
Combine beans, snap peas, green peas, mint, and prosciutto in a large bowl. Combine vinegar and remaining ingredients in a small bowl; stir with a whisk. Pour over bean mixture, and toss well.