Lay the cucumber slices on kitchen paper to dry. Whizz the cheese, mint and some seasoning in a small food processor or with a hand blender. Stir in most of the chives, then spread the cheese mixture over 6 slices of bread. Reserve 12 slices of cucumber, and arrange the rest over the cheese.
Layer up 2 slices of cucumber-topped bread, then top with a plain slice. Repeat this twice more. Use a good knife to trim the crusts off the tripledeckers, then cut each into 4 squares. Top each mini sandwich with another cucumber slice. Sprinkle with the remaining chives to serve.