Heat the broiler and place the whole pepper on a foil-lined sheet 6 inches below the heat. Broil for 10 to 14 minutes, turning until blistered on all sides.
Cover the pepper in a bowl for 10 minutes, then peel, remove seeds and blot dry. Dice finely and spread on a plate to cool completely for about 10 minutes.
Stir mayonnaise, yogurt, onion powder, cayenne and black pepper together. Fold in grated white cheddar and cooled pepper dice, keeping the cheese visibly shredded.
Cover and refrigerate for 30 minutes until firm and cold; cooling and chilling are included in prep time.
Spread one-quarter of the cheese mixture evenly over each of 4 rye slices, close with remaining bread and cut diagonally. Serve cold or refrigerate at 40°F or below, packing with ice packs for transport.