In a large saucepan, bring the coconut milk and curry paste to a boil over medium-high heat, whisking constantly until smooth, about 2 minutes. Add the chicken broth, carrot, red bell pepper, onion, baby corn, basil, chile, lime leaves and fish sauce. Bring the mixture to a simmer over medium-low heat. Cover the pan and cook until the vegetables are tender, about 20 minutes.
Pour the oil into a large wide saucepan. Attach a deep-fry thermometer to the side of the saucepan and heat the oil to 350 degrees F. Add half of the noodles and fry until crisp, about 20 seconds. Drain on paper towels and set aside.
Remove the lid from the curry and add the shrimp, snap peas and the remaining noodles. Simmer, uncovered, until the shrimp is cooked through, 5 to 7 minutes. Remove the lime leaves and basil sprigs and discard.
Ladle the curry into bowls. Garnish with the fried noodles, cilantro and peanuts.
To substitute dried rice noodles, use 4 ounces and soak half in water as directed; use the rest for frying.