Sauté the mushrooms and garlic over medium heat in the 2 tbsp. water until tender; cover between stirring to keep them from drying out. Remove from heat and add the spaghetti sauce.
Place the tofu and thawed spinach in the food processor and process briefly. Add the remaining filling ingredients to the processor and blend until smooth.
Preheat the oven to 375 degrees.
Spread half of the sauce in the bottom of a 9×12-inch pan. Place a layer of noodles over the sauce, using three dry noodles and leaving a little space in between them.
Spread half of the tofu mixture on the noodles (you can drop it by spoonfuls and then spread it or hold each noodle in the palm of your hand and spread the tofu on it before placing it in the pan).
Cover with another layer of 3 noodles and then spread the remaining tofu mixture over them. Top with a final layer of noodles, and pour the remaining sauce over this.
Cover the dish tightly with foil, and bake for 30 minutes. Then, remove the foil and bake for another 30 minutes.
Remove from the oven and sprinkle with vegan Parmesan or cheese and sliced black olives if you want. The lasagna will cut better if you allow it to cool for 15 minutes before serving.
Optional high-fat version: Sprinkle with a grated vegan cheese during the last 5 minutes of cooking.