To press tofu, wrap tofu in paper towels and place on a plate. Place a flat heavy object on top of the tofu, such as a book or cutting board. Let the tofu sit for at least 30 minutes to drain out some of the water. Alternatively, you can use a tofu press*.
Mix miso sauce ingredients together in a small bowl and set aside.
Pat tofu dry with a towel. Heat 1 tbsp oil in a deep pan or wok over medium heat and cook tofu until browned on both sides, about 3-5 minutes each side. Transfer tofu to a plate.
Add remaining 1 tsp oil to the wok and heat over medium. Add ginger and cook 30-60 seconds until fragrant.
Add napa cabbage and cook until tender, about 3 minutes.
Add mushrooms and cook until they soften, about 2 minutes.
Reduce heat to medium low. Add tofu and pour the sauce over. Simmer for 3 minutes, stirring frequently and spooning the sauce over the tofu.
Serve immediately or cover and let the tofu soak in the sauce until ready to serve.