Combine all ingredients except cornstarch and green onions in a large pot and bring to a boil over high heat. Reduce to low and simmer, covered, turning occasionally, until chicken is tender, about 30 to 35 minutes more.
Remove chicken to a serving platter. Remove garlic and ginger and discard. Bring sauce to a boil, skim off excess fat, and cook until reduced slightly, about 10 minutes. Whisk in cornstarch mixture and bring to a boil. Remove from heat, add chicken, turn to coat, and serve chicken with sauce and sliced green onions.