Warm a large saucepan over medium heat. Add oil; swirl to coat. Add lemongrass, shallot and ginger; cook, stirring, until fragrant and beginning to soften, about 2 minutes. Add broth, fish sauce and sugar; bring to a boil, then reduce heat to medium-low and simmer for 10 minutes. Strain, pressing on solids; discard solids.
Return soup to pan; bring back to a simmer over medium-low heat. Stir in mushrooms and bell pepper; cook for 2 minutes. Stir in chicken; cook until just opaque, about 2 minutes longer. Stir in coconut milk and lime juice; cook, stirring occasionally, until warmed through. Ladle into serving bowls and top with cilantro, scallions and chilies, if desired. Serve with lime wedges.