In 2 batches in a food processor, puree the melon, mint, 2 1/2 tablespoons of the liqueur, lime juice and sugar. Place in a pitcher or bowl and refrigerate until well chilled.
In a bowl, toss the honeydew balls with the remaining 1/2 tablespoon liqueur and let macerate in the refrigerator.
Cut each breadstick in half crosswise and wrap each piece with a slice of ham.
To serve, ladle the soup into 4 shallow bowls. Swirl 1 1/2 teaspoons of the crème fraiche into the center of each serving and arrange a portion of cantaloupe decoratively in each bowl. Arrange 2 breadsticks on the rim of each bowl, and serve.
