Several hours or day before serving, prepare Teriyaki Marinade: In small jar with tight-fitting lid, whisk together soy sauce, sugar, lemon juice, sesame oil, and ginger; cover and set aside until ready to use.
Several hours before serving, prepare salmon mixture for Burgers: In food processor fitted with chopping blade, process salmon, green onion, garlic, gingerroot, soy sauce, and sesame oil until well combined. Divide into 4 equal parts; shape each into a 5-inch round burger. Cover and refrigerate at least 1 hour before grilling. (Burgers must be refrigerated for at least the recommeded time to maintain shape during cooking.) About 30 minutes before grilling, heat indoor or outdoor grill to medium. In medium-size bowl, combine pineapple rings, green onions, and Teriyaki Marinade; cover and refrigerate to 10 minutes. Drain marinade into serving pitcher and set aside.
Coat grill rack with nonstick cooking spray. Grill burgers 6 inches above heat until cooked through, turning once--about 10 minutes. Grill pineapple rings and green onions until heated through, turning once--about 1 to 2 minutes. Grill rolls just until lightly toasted.
On 4 individual plates, assemble burgers: Placed salmon burgers on bottom halves of rolls. Top each with a pineapple slice and 3 green onion halves; cover with tops of rolls. Serve with reserved Teriyaki Marinade.