Halve the cucumber lengthwise and use a small spoon to scoop out the seeds. Use a mandoline to slice the cucumber into thin lengthwise strips.

Fold the chives into the whipped cream cheese, and spread onto one side of each piece of bread.

Layer the cucumber slices onto each piece of bread, overlapping slightly.

Use a very sharp knife to trim the edges of the cucumber and slice the crust off of the bread. Slice each piece of bread in half on the diagonal to form triangles.

Top each sandwich with more chives and serve right away. If the sandwiches need to be stored, wrap in lightly dampened paper towels or lightly dampened dish cloth and refrigerate.