Heat the oil in a small pot over medium heat. Add the scallion whites, gochugaru, gochujang, and chopped kimchi. Cook for about 1 minute, stirring frequently, until fragrant.
Pour in the chicken bone broth and bring to a gentle simmer. Add the instant noodles.
Arrange the sliced shiitake mushrooms, soft tofu, additional kimchi, and egg over the noodles.
Cover the pot and cook until the noodles are tender, the mushrooms have softened, and the egg is cooked to your desired doneness.
Remove from the heat and serve immediately while hot.