Peel the cucumber and slice it very, very thin. Lay the cucumber slices out on a sheet of paper towel and cover them with a second sheet. Let stand until the paper towels soak up the cucumber slices' liquid, about 15 minutes.
Mash the butter, scallions, dill, horseradish, salt and pepper together in a small bowl until well blended. Spread 5 of the bread slices with half of the dill butter, then top the butter with a layer of cucumber slices—it's fine if they overlap a little bit. Spread the remaining butter over the remaining bread and top the sandwiches. Cut each sandwich into 4 pieces.