Heat gas or charcoal grill. Cut 1 burger bun into 1/2-inch cubes (1 1/2 cups); place in medium bowl. Add lemon peel, lemon juice and egg white; mix well. Stir in salmon. Shape mixture into 5 flat oval-shaped patties, 1/2 inch thick.
In small bowl, mix mayonnaise and dill; set aside.
When grill is heated, place patties on gas grill over medium-high heat or on charcoal grill over medium-high coals; cover grill. Cook 8 to 16 minutes, turning once, until patties are lightly browned on both sides.
To serve, spread cut sides of bun halves with mayonnaise mixture. Place patties on bottom halves of buns. Top with tomato and cucumber slices. Cover with top halves of buns.