Place chilled ham cubes in a food processor and pulse 6 to 8 times until finely chopped but not a smooth paste. Transfer to a bowl.
Stir mayonnaise, drained relish, Dijon, paprika and pepper together in a small bowl, then fold into the ham with diced celery.
Cover and refrigerate for 20 minutes so the filling firms; chilling is included in prep time.
Lay one dry lettuce leaf over each of 4 rye slices and spread one-quarter of the ham salad evenly over it. Close with remaining bread and cut each sandwich in half.
Serve cold or keep refrigerated at 40°F or below, transporting in an insulated bag with ice packs.