Put lemongrass, ginger and broth in a large pot. Bring to a gentle simmer and cook for 10 minutes, then remove the lemongrass and ginger with a slotted spoon.
Stir in coconut milk and return to a gentle simmer. Add chicken, mushrooms and sliced chili.
Cook for 7 to 10 minutes, stirring gently, until the chicken is opaque and reaches 165°F and the mushrooms are tender.
Stir in fish sauce and sugar and heat for 1 minute to dissolve the sugar.
Remove from the heat and stir in lime juice. Divide the chicken, mushrooms and coconut broth among bowls.