Heat 1 teaspoon sesame oil in a large pot over low heat; cook and stir fresh ginger in the hot oil for 10 minutes. Pour chicken stock over ginger, cover pot, and simmer for 30 minutes.
Stir red onion and carrots into stock and simmer until carrots are tender, about 10 minutes. Gradually add curry powder, ground ginger, cayenne pepper, salt and black pepper to the stock; stir in jalapeno pepper and half the lime juice. Simmer soup over low heat, stirring about every 15 minutes, until flavours have blended, about 1 hour.
Fill a large pot with lightly salted water and bring to the boil. Drop in noodles, a few at a time and return to the boil. Cook uncovered, stirring occasionally, until the noodles are cooked through, but still firm to the bite, 10 to 12 minutes; drain.
Heat 1 teaspoon sesame oil in a pan over medium heat; cook and stir chicken in the hot oil until cooked through, about 10 minutes. Transfer chicken to the soup; stir in sliced leek and spring onion. Simmer for 5 to 10 more minutes.
Portion noodles into each serving bowl; ladle stock over noodles. Add remaining lime juice to each serving.