Peel the cucumber, then cut in half lengthwise. Scoop out seeds and discard. Cut cucumber into 1/2-inch-thick slices.
In a blender container or food processor combine cucumber, yogurt, lime juice, honey, cumin, and salt. Cover and blend or process until smooth. Blend in milk, if desired. Stir in snipped mint. Cover and chill 2 to 24 hours. Stir before serving. Garnish with fresh mint sprigs. Makes 4 side-dish servings.