In a medium bowl stir together mayonnaise dressing, mustard, dill, and pepper. Stir in tuna and carrot; set aside.
Use a fork to hollow out the tops and bottoms of frankfurter buns, leaving 1/4-inch shells. Sprinkle cheese into hollowed-out bun bottoms. Spoon tuna mixture over cheese. Add bun tops. Wrap tightly in plastic wrap. Chill for 2 to 24 hours. Pack in insulated containers with ice packs. Makes 4 sandwiches.