In a large saucepan, bring the coconut milk, chicken broth and curry paste to a boil over medium-high heat, whisking until smooth, about 2 minutes. Add the carrot, red pepper, onion, corn, chile, basil, lime leaves and fish sauce. Bring to a simmer, cover the pan and cook for 30 minutes, or until the vegetables are tender.
Add the shrimp, snap peas and half of the noodles (reserve the remaining noodles for frying). Simmer, uncovered, until the shrimp are cooked through, 5 to 7 minutes. Remove and discard the lime leaves and basil sprigs.
Pour the canola oil into a large, wide saucepan fitted with a deep-fry thermometer. Heat the oil to 350 degrees F. Add the reserved noodles and fry them until crispy, about 20 seconds. Drain on paper towels.
Ladle the curry into four bowls. Garnish with cilantro and peanuts and serve.