Use a vegetable peeler to cut cucumber lengthwise into broad thin ribbons, stopping when you reach the watery seed center; reserve the center for another use. Lay ribbons on paper towels, sprinkle with salt and let stand for 10 minutes.
Blot cucumber ribbons firmly with fresh towels until no beads of water remain. Stir labneh with chopped mint and black pepper.
Spread the labneh mixture evenly over one face of all 8 bread slices. Fold and overlap cucumber ribbons in thin layers over 4 slices.
Close with the remaining bread, spread-side down. Trim the crusts neatly and cut each sandwich into 2 rectangles.
Serve 2 rectangles per portion or refrigerate promptly at 40°F or below, packing in a rigid container with ice packs for transport.