Peel the cucumber. Split into halves vertically. Scrape off the seeds with a teaspoon. Cut the flesh into small cubes.
Cut the mangoes, remove the stones and discard. Scoop out the flesh and cut into small cubes.
Put the cucumber and mango cubes into the blender. Pour in the lime or lemon juice. Add the simple syrup or sugar (start with a tablespoonful; you can add more later, if needed).
Add about half a cup of crushed ice into the blender. Process the mixture until the cucumber and mangoes are pureed. Taste. Add more syrup or sugar, if needed.
Tear the mint leaves and throw into the blender. Blitz a few times to chop the leaves.
Pour the smoothie into glasses. Garnish with whole mint leaves. Serve cold.