Combine the soy sauce, vinegar, sugar, ginger, and chicken broth in a medium saucepan and bring to a boil. Add the chicken and carrots and simmer for 6 minutes. Stir in the Brussels sprouts and continue cooking until the chicken is cooked through, about 2 minutes more. Divide among individual bowls and top with the scallions.
Upgrade: Add more flavor to this dish by drizzling it with aromatic toasted sesame oil just before serving.