Put the shiitakes in a bowl and cover with about 1-1/2-cups boiling water. Set a small plate or pot lid over the mushrooms to ensure they’re submerged, and soak until softened, about 20 min. Discard the mushroom stems and thinly slice the caps. Set a sieve lined with cheesecloth or a damp paper towel over a measuring cup and strain the soaking liquid through it. Reserve the liquid and shiitakes separately.
Rinse the chicken thighs and pat dry with paper towels; season with salt. Heat the oil in a heavy soup pot or Dutch oven over medium-low heat. When the oil is hot, add the chicken, cover the pot, and cook for about 10 min, turning once or twice. The aim is to sweat (not brown) the chicken until it loses its raw pink color on the outside and begins to exude some of its juices. Stir in the leeks, ginger, and sliced shiitakes; cover and cook until the leeks begin to soften, about another 5 min.
Add the broth, water, reserved mushroom soaking liquid, mirin, soy sauce, vinegar, and chile paste, and bring to a vigorous simmer over high heat. Adjust the heat to maintain a gentle simmer and cook uncovered until the chicken is very tender and falling off the bone, about 45 min. Remove the chicken and set it aside.
Meanwhile, bring a large pot of well-salted water to a boil. Cook the noodles until just tender, about 3 min. Drain and rinse under cold water.
When the chicken is cool enough to handle, remove the meat from the bones and roughly chop or tear it into bite-size pieces. Return the meat to the soup. Add the sesame oil and season the soup to taste with salt and pepper.
Stir the noodles into the soup and let them heat through. Ladle the soup into bowls, top with the sliced scallions, and serve immediately.