Bring a large pot of salted water to a boil. Add tofu cubes and simmer until firm and puffed, about 3 to 4 minutes. Using a slotted spoon, carefully remove tofu from pot and drain on a paper towel–lined plate. Empty pot.
Return pot to stove and pour in vegetable oil. Heat over medium-high, and add garlic, ginger, and light green and white parts of scallions. Cook, stirring, until softened, 1 to 2 minutes. Stir in chopped kimchi, gochujang, and gochugaru and cook, stirring often, until ingredients begin to brown, 5 to 8 minutes.
Add 8 cups water and reserved kimchi liquid to pot and bring to a boil. Reduce heat and simmer for 35 minutes, until kimchi is translucent and broth is a rich red color. Taste and add salt as needed.
While stew simmers, spiralize the beet, carrot, and turnip into noodles, using the crude shredding blade.
Add soy sauce and tofu to the soup and simmer for 10 minutes. Stir in the vegetable noodles and simmer until the noodles are cooked through but still firm, 3 to 4 minutes.
Divide soup between bowls and garnish with scallion greens, sesame oil, and sesame seeds.