Wrap tofu in a paper towel; cook in microwave for 1 minute. Squeeze extra water from tofu and wrap in a dry paper towel. Let sit to drain extra water, about 5 minutes. Remove paper towel and cut tofu into cubes.
Combine luncheon meat, garlic, and ginger in a bowl. Whisk sake, miso paste, soy sauce, and sugar in a separate bowl.
Heat 2 teaspoons vegetable oil in a skillet over medium heat; cook and stir eggs until scrambled and cooked through, about 5 minutes. Transfer eggs to a plate. Cook and stir tofu in the same skillet until browned on all sides, 5 to 10 minutes. Transfer tofu to plate with eggs. Cook and stir luncheon meat mixture in the same skillet until cooked through and garlic is lightly browned, about 5 minutes.
Heat 2 teaspoons vegetable oil in a separate skillet over medium heat; cook and stir onion, cabbage, carrots, and mushrooms until onions are translucent and cabbage is softened, 10 to 12 minutes. Add eggs, tofu, luncheon meat, and sake mixture to vegetable mixture; stir to coat. Cook until heated through, about 1 minute. Garnish with green onion.