Add the scallions and oil to a large pan. Turn on the heat to medium, and sauté for 5 minutes, stirring occasionally.
Add the shallot and yellow onion. Stir to combine and cook for 2 minutes, stirring as needed.
Add the kimchi and mushrooms. Stir to combine and cook for 3 minutes, stirring as needed.
Add the soy sauce, mirin and sugar. Stir to combine and cook for 3 minutes, stirring as needed.
Add the Korean chili powder and stir to combine. Turn off the heat.
Add the rice and mix until very well combined.
Turn heat up to medium-high. With a spatula, even out the rice mixture so that it is flat. Sprinkle the cheese and corn on top, and cover for 3 minutes.
To plate, add the kimchi rice to a bowl and sprinkle with sesame seeds if using. Add the shredded seaweed to the center of the rice, and 2 pieces of halved seaweed to the side of the bowl. Carefully add the pickeled eggs if using, and enjoy!