Thinly slice the cucumber on a mandoline and place in a colander over a bowl. Season with 1/2 teaspoon salt and allow the cucumbers to sit for 5 minutes. Squeeze out the excess water and place the cucumbers in a clean medium bowl. Add the sour cream, dill and 1/4 teaspoon salt and stir with a rubber spatula to combine.
Lay the smoked salmon on one half of a platter lightly, so there is movement to the plate, leaving open the other half of the platter. Pile the cucumber mixture on a quarter of the platter or spoon into a bowl that fits on the platter. Serve the toasted pumpernickel bread on the other quarter of the platter.