Bring a large pot of salted water to a boil. Heat the coconut oil in a large saucepan over medium heat. Stir in the curry paste, fish sauce and brown sugar and cook, stirring, until the mixture darkens and starts sticking to the pan, about 1 minute. Add the chicken broth and 3 cups water. Bring to a boil, then reduce the heat and simmer 5 minutes.
Meanwhile, halve the cherry tomatoes. Cut the bok choy into 1-inch pieces and chop half the cilantro.
Add the rice noodles to the pot of boiling water and cook as the label directs. Drain the noodles and divide among bowls.
Add the lime juice to the broth mixture; season with salt. Stir in the tomatoes and cook until softened, about 1 minute. Add the shrimp, bok choy and chopped cilantro. Cook until the shrimp is opaque, 1 1/2 to 2 minutes. Ladle the soup over the noodles and top with the remaining cilantro.