Place chicken and 1/2 cup green onions in a medium saucepan; cover with water. Bring to a boil over medium-high heat; cover, reduce heat, and simmer 5 minutes. Remove from heat; let stand 10 minutes. Drain and discard green onions. Place chicken on a work surface or cutting board, and shred chicken with 2 forks.
Combine juice, sugar, and garlic in a small microwave-safe bowl. Cover with plastic wrap; microwave at HIGH for 20 seconds or until sugar melts. Cool to room temperature; stir in fish sauce.
Combine remaining 1 cup onions, chicken, cabbage, and remaining ingredients in a large bowl. Drizzle with juice mixture; toss well to coat. Cover and let stand 5 minutes before serving.