Warm oil in soup pot over medium-high heat; add tofu and sear until golden on one side, 2 min. Add mushrooms and cook 1 min more. Add garlic, white part of chopped scallions, and ginger and cook until fragrant, 1 min. Add brown sugar, tamari, gochujang, vegetable broth, kimchi and kimchi juice, bringing to a boil over high heat. Serve immediately in four bowls, breaking an egg over each and garnishing with the green parts of the chopped green onion. Serve with rice, more kimchi and more gochujang on the side, as desired.