Wash the large cucumber under the tap and pat dry. Peeling is not necessary. Using a vegetable peeler or mandoline, shave thin ribbons lengthways from the cucumber. Turn the cucumber while shaving and stop once you reach the watery seed core. The centre is not used. Put the shaved cucumber ribbons in a large bowl.
Whisk together with a fork or whisk a smooth dressing from the extra virgin olive oil, fresh lemon juice, sea salt, freshly ground black pepper and two-thirds of the finely chopped fresh dill. Pour this dressing over the cucumber ribbons. Gently toss everything so that the ribbons are coated all over, but take care not to break them. Let the cucumber marinate at room temperature for ten minutes.
Arrange the marinated cucumber ribbons as a light bed on a large serving plate or on individual plates. Separate the thin slices of smoked salmon from each other. Arrange the salmon over the plate and drape loosely over and between the cucumber ribbons. Sprinkle the remaining fresh dill over everything to finish the plates.
Place the plate or plates directly on the table and serve the dish.