Warm chicken broth and coconut milk together in a stockpot over medium heat; add chile sauce, lime juice, fish sauce, brown sugar, ginger, and kaffir lime leaves. Bring broth mixture to a low simmer until flavors have blended, about 15 minutes.
Remove stockpot from heat and strain out ginger and lime leaves. Return broth mixture to the stockpot and bring to a simmer. Cook chicken in the simmering broth until no longer pink in the center, about 5 minutes. Add mushrooms, onion, zucchini, lemongrass, and basil and simmer until onion softens, about 3 minutes.
