Using the back of a knife, lightly smash lemongrass and ginger; cut lemongrass into 4†pieces. Bring lemongrass, ginger, lime leaves, and broth to a boil in a large saucepan. Reduce heat and simmer until flavors are melded, 8–10 minutes. Strain broth into clean saucepan; discard solids. Add chicken and return to a boil. Reduce heat, add mushrooms, and simmer, skimming occasionally, until chicken is cooked through and mushrooms are soft, 20–25 minutes. Mix in coconut milk, fish sauce, and sugar. Divide soup among bowls. Serve with chili oil, cilantro, and lime wedges.