Add chicken broth and coconut milk to a stock pot and beging to warm. Add fish sauce, brown sugar, lime juice, red chili sauce, ginger, and kaffir lime leaves to pot and bring up to a low simmer for around 15 minutes to infuse the broth with the flavors (avoid a rapid boil).
Remove pot from burner and strain out ginger and lime leaves, add remaining soup back to the pot and return to burner.
Add sliced chicken and cook for 5 minutes.
Add lemongrass, mushrooms, onion, zucchini and basil.