In a 5-quart pot, heat the oil over medium-high heat. Add the mushrooms and cook, stirring occasionally, until browned, about 5 minutes. Add the ginger and cook, stirring, until fragrant, about 1 minute. Add the chicken broth, mirin, and 2 cups water; bring to a simmer. Add the chicken, lower the heat to medium, and simmer gently until the chicken is cooked through, about 15 minutes.
Transfer the chicken to a cutting board and chop it or shred it with two forks into bite-size pieces. Return the chicken to the pot. Turn off the heat and whisk in the kimchi and miso. Serve topped with the scallions.