Warm a large skillet over medium heat. Add coconut milk, curry paste, and turmeric. Stir well, bring to a boil, then reduce heat and simmer for 5 minutes.
Add broccoli, carrots, snowpeas, chickpeas and chile, if using. Cover and cook for 8 to 10 minutes, or until vegetables are soft, stirring once or twice. Stir in lime juice and season with salt and pepper.
Serve curry over rice, cauliflower rice or noodles, if desired, garnished with fresh basil.
