Place cucumber slices between layers of paper towels to remove excess moisture.
In a small bowl, combine the mint, butter, and cream cheese; spread on one side of each slice of bread. Lay the cucumber slices onto the buttered side of 8 slices of bread. Sprinkle with salt. Top with the remaining slices of bread, buttered side down.
Gently cut the crusts from each sandwich with a sharp knife. Cut the sandwiches in half diagonally and then cut in half again.
Makes 8 whole sandwiches, or 16 halves, or 32 fourths.