Combine the first 4 ingredients in a large bowl. Divide the turkey mixture into 4 equal portions, shaping each portion into a 1/2-inch-thick patty. Combine the teriyaki sauce and water in a small bowl.
Place a large nonstick skillet coated with cooking spray over medium heat until hot. Add the onion slices; cover and cook 10 minutes or until onion is golden brown, stirring frequently. Stir in 1 tablespoon teriyaki mixture. Remove onion from pan, and keep warm. Add olive oil to pan. Add patties, and cook 5 minutes over medium heat. Add 3 tablespoons teriyaki mixture to pan. Carefully turn patties over, and cook for 3 minutes or until golden. Place 1 patty on bottom half of each bun, and top each patty with 1/4 cup onion, 2 tomato slices, and 1 lettuce leaf. Top with remaining bun halves.
Note: You can freeze any uncooked turkey patties for up to 1 month. To freeze, separate the patties with wax paper; place them in a heavy-duty zip-top plastic bag, remove excess air, seal, and freeze. Thaw in the refrigerator.