Peel cucumber lengthwise into broad, thin ribbons, stopping at watery seed center; reserve center for another use. Salt ribbons on paper towels; rest 10 minutes.
Blot firmly with fresh towels until no water beads remain. Mix labneh, mint, and pepper.
Spread labneh evenly on one face of all 8 slices. Fold and overlap cucumber thinly over 4 slices.
Close with remaining bread, spread-side down. Neatly trim crusts; cut each into 2 rectangles.
Serve 2 rectangles per portion or promptly refrigerate at 40°F or below; transport in a rigid container with ice packs.