Combine shiitake mushrooms and 3 cups of the chicken stock in a small saucepan and bring to a simmer. Remove from heat and let stand until mushrooms are tender and broth is flavorful, about 30 minutes. Remove mushrooms and discard stems. Thinly slice caps and reserve. Strain broth through a fine mesh sieve, combine with the remaining chicken stock and set aside.

In a medium saucepan heat the vegetable oil and, when hot, add the ginger, garlic and crushed red pepper; cook for 2 minutes, or until fragrant. Add the chicken stock and reserved shiitakes and bring to a simmer. Cook for 30 minutes, or until broth is flavorful. Add the chicken or tofu and stir to combine. In a small bowl or cup, combine the lime juice and cornstarch and stir until smooth. Whisk the cornstarch mixture into the hot soup and bring to a boil. Reduce heat to a simmer and cook for 5 minutes, until thickened. Add the soy sauce and sesame oil and adjust the seasoning to taste. Serve the soup, drizzled with hot chile oil to taste, and garnished with some of the sliced scallions.