Preheat your oven to 375°F/190°C. Heat 2 tablespoons of oil in a skillet over medium-high heat.
Add the onion, and cook for 5 minutes, or until it begins to turn translucent. Add the mushrooms and cook for a few more minutes until tender.
Turn off the heat, and mix in the kimchi, kimchi juice, cooked rice, and gochujang until well-combined. Transfer to a small baking dish. Top evenly with shredded cheese. Bake for 12 - 15 minutes, until the cheese is lightly browned and bubbly.
While the rice is baking, fry the eggs in 1 tablespoon of oil to serve with your kimchi rice bake. Garnish with chopped scallion and enjoy!