In a 4-quart Dutch oven cook and stir ginger and garlic in hot oil over medium heat for 30 seconds. Add onion; cook and stir for 4 to 6 minutes or until onion is tender but not brown. Stir in broth; bring to boiling. Add chicken, carrots, lemon peel, and crushed red pepper. Return to boiling; reduce heat. Cover and simmer for 15 to 20 minutes or until chicken and carrots are tender.
Stir coconut milk, sweet pepper, and mushrooms into soup; heat through. Top each serving with peanuts and cilantro. If desired, serve with lime wedges to squeeze over soup.