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. 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. Add the soy sauce and sesame oil and serve the soup, drizzled with hot chile oil to taste, and garnished with some of the sliced scallions.
