In a medium bowl, combine honeydew, cantaloupe, 2 tablespoons of sugar, lime zest and juice, and slivered mint. Refrigerate 15 minutes, tossing occasionally, until sugar dissolves and mixture is well chilled.
When ready to serve, spread remaining tablespoon of sugar onto a flat plate. Pour a little water into a shallow bowl and dip rim of each of 4 wine or martini glasses into water; press wet rims into sugar to coat edge of each glass. Spoon 1 cup of fruit mixture into each glass; garnish with mint sprigs. Yields 1 cup per serving.