Make the Tofu Ricotta and Garlic Béchamel Sauce (recipes below).
Preheat the oven to 350F. Adjust the oven rack to center position.
Soak the lasagna noodles. Arrange the lasagna noodles in a 9- by 13-inch baking pan and cover with hot (not boiling) water. Let soak for 10 minutes, then remove them and arrange in a single layer in an oiled baking sheet. Pat dry the 9- by 13-inch baking pan.
Chop the vegetables. Working in batches, place the mushrooms in a food processor bowl fitted with the S-blade and pulse a few times until the mushrooms are roughly chopped. For the Brussels sprouts, use the slicing attachment to grate the sprouts (or buy shredded Brussels sprouts).
Heat 1 tablespoon of olive oil in a large (12-inch) skillet over medium-high heat until shimmering. Add the shredded Brussels sprouts and 1/2 teaspoon kosher salt and black pepper to taste. Cook until the sprouts are still bright green in color, stirring occasionally, for 6 minutes. Stir in the balsamic vinegar and cook for 1 minute. Remove the sprouts and set them aside in a bowl. Clean out the skillet.
Using the same skillet, add the vegan butter until melted. Add the mushrooms, stirring occasionally, until the moisture evaporates and they begin to brown, about 8-10 minutes. Once starting to brown, add the garlic, thyme, and rosemary, 1/2 teaspoon kosher salt, and black pepper, and cook for 1 minute. Push the mushrooms to one side and add the miso paste. Once the miso has warmed and softened, stir it into the mushrooms. Transfer to a separate bowl than the Brussels.
Assemble the lasagna: Pour 1/3 cup béchamel onto the bottom of the 9- by 13-inch pan to cover. Add a layer of lasagna sheets (3-4 sheets), 1/3 of the sprouts, 1/3 of the mushrooms, and 1/3 of the remaining béchamel on top, making sure to cover all of the lasagna sheets with sauce. Add 1/3 of the tofu ricotta, spreading it out as evenly as you can. Repeat the layers two times. Finish the top with the shredded vegan mozzarella.
Cover the lasagna with aluminum foil and bake for 30 minutes. Remove foil and bake for another 15 minutes. If desired, place the lasagna under the broiler for a few minutes, keeping a close eye on it to prevent burning.