Bring water and 1 teaspoon salt to a boil in a large pot. Add chicken legs and simmer until no longer pink in the center and juices run clear, about 10 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Reserving the liquid, remove chicken and allow to cool, then remove meat from the bones and shred with two forks. Discard the skin and bones.

Meanwhile, place shiitake mushrooms in a bowl and pour enough warm water over them to cover completely; allow to soak until pliable, about 30 minutes. Remove mushrooms, reserving water; slice, and set aside. Place noodles in reserved water, adding more warm water if needed to cover; allow to soak until soft, about 10 minutes. Drain and cut noodles if desired.

Heat olive oil in a skillet over medium heat. Add onion and garlic; cook and stir until softened, about 5 minutes. Add achiote powder and continue to cook and stir until the mixture is well coated. Stir in shredded chicken meat, sliced shiitake mushrooms, and fish sauce; season with salt and pepper.

Stir in chicken broth and reserved cooking liquid from chicken. Bring to a boil for 5 minutes. Add noodles and cook 5 minutes more. Garnish with green onions.