1. Bring 3 cans broth to a boil in a pot with ginger and garlic. Add chicken, cover, and remove from heat. Let sit, stirring occasionally, until chicken is cooked, 15 minutes.
2. Using a slotted spoon, transfer chicken to a serving bowl. Add remaining 1 can broth to pot and bring to a boil. Serve broth and chicken in separate bowls.
Make ahead: Up to 1 day. Bring all broth to a boil with chicken, then remove chicken and chill separately from broth.