In a food processor combine salmon, egg, green onions, 1 tablespoon oregano, half of the lemon peel, bread slice, 1/4 teaspoon of the salt, and 1/8 teaspoon of the pepper. Cover and pulse 5 times or until ingredients are combined and mixture holds together, scraping sides as needed. Shape mixture into 4 patties, each about 3/4-inch thick. Place on plate; cover and chill for 30 minutes.
In a small bowl stir together the yogurt, cucumber, remaining lemon peel, lemon juice, remaining 1/4 teaspoon salt, and remaining 1/8 teaspoon pepper. Cover and chill until serving time.
For charcoal grill, arrange medium-hot coals around edge of grill. Place the plank on the grill rack directly over the coals for 5 minutes or until plank smokes and crackles. Arrange salmon patties on plank. Place plank in center of grill rack. Cover and grill for 18 to 20 minutes or until an instant-read thermometer inserted in patties registers 160 degrees F. (For gas grill, preheat grill. Reduce heat to medium. Adjust heat for indirect cooking. Place plank on grill rack over heat for 5 minutes or until smoking and crackling. Arrange patties on plank. Place plank over unlit side of grill and grill as above.)
Serve salmon on toasted buns with lettuce leaves and yogurt mixture. Top with additional oregano leaves and additional pepper, if desired.