Heat oil in a large skillet or wok over high heat; add shallots and ginger. Stir-fry until fragrant, about 1 minute. Reduce heat to medium; stir in curry paste. Stir in chicken broth; cook until broth is reduced by half, about 10 minutes.
Stir in chicken, coconut milk, bamboo shoots and salt; heat to a boil. Reduce heat; simmer 5 minutes. Stir in basil.
Serve in bowls over steamed rice or noodles with lime wedges on the side for squeezing over.