Combine first 6 ingredients in a large bowl, stirring with a whisk. Gradually add oil, stirring constantly with a whisk.
Remove and discard seeds from cantaloupe and honeydew melon. Cut melons into 2-inch-wide wedges; remove rinds. Cut melon wedges into long, thin ribbons using a mandoline. Add melon ribbons to dressing; toss gently to coat. Sprinkle with mint. Serve immediately.
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