Heat 1 teaspoon sesame oil in a large pot over low heat; cook and stir ginger in the hot oil for 10 minutes. Pour chicken broth over ginger, cover pot, and simmer for 30 minutes.

Stir onion and carrots into broth and simmer until carrots are tender, about 10 minutes. Gradually add curry powder, ground ginger, cayenne pepper, salt, and black pepper to broth; stir in jalapeno pepper and half the lime juice. Simmer broth over low heat, stirring about every 15 minutes, until flavors have blended, about 1 hour.

Fill a large pot with lightly salted water and bring to a rolling boil. Drop udon in a few noodles at a time and return to a boil. Cook uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, 10 to 12 minutes; drain.

Heat 1 teaspoon sesame oil in a skillet over medium heat; cook and stir chicken in the hot oil until no longer pink in the center, about 10 minutes. Transfer chicken to broth and stir leek and green onion into broth; simmer for 5 to 10 more minutes. Remove pot from heat.

Portion udon noodles into each serving bowl; ladle broth over noodles. Add remaining lime juice to each serving.