In a large saucepan, bring ginger, lime leaves, lemongrass (smash pieces with back of large knive) and broth to a boil. Then reduce heat and simmer until flavors meld together, about 10-15 minutes. Strain broth into a large bowl, discard solids. Wipe out saucepan, and pour broth back into it.
Add chicken to the broth, and bring to a boil. Reduce heat, add mushrooms and simmer. Stir occasionally, skimming any fat off the top, until chicken is cooked through, about 20-25 minutes. Mix in coconut milk, fish sauce and sugar, and simmer for another 5 minutes.
Pour soup into bowls and serve with chili oil, cilantro and lime wedges.