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In a large bowl, cover the rice vermicelli with cold water and let soak until pliable, about 20 minutes. Drain the vermicelli.
Heat the vegetable oil in a wok or large saucepan. Add the Naam Ya Chile Paste and stir-fry over moderately high heat until fragrant, about 4 minutes. Add the thin coconut milk and bring to a simmer. Add the thick coconut milk, fish sauce, palm sugar and crabmeat and simmer until the curry is heated through, about 2 minutes.
Meanwhile, bring a large saucepan of water to a boil. Add the rice vermicelli and cook, stirring, until just tender, about 2 minutes. Drain the vermicelli and transfer to a large shallow serving bowl. Spoon the crab curry on top and garnish with the bean sprouts and basil leaves.
Dried rice vermicelli, thin and thick unsweetened coconut milk, Thai fish sauce and palm sugar are available at Asian markets, well-stocked supermarkets and specialty food stores.