Preheat the oven to 350 degrees F.
Layer the eggplant in a 13 x 9-inch baking dish. Pour 1/2 cup of the broth over the top and sprinkle with the rosemary. Cover with foil and bake for 30 minutes.
Line a rimmed baking sheet with parchment paper and arrange the mushrooms on it with the gills facing up. Sprinkle with the remaining 1/2 cup of broth and the shallot and bake for 20 minutes. Let cool. When cool to the touch, cut the mushrooms on a diagonal into wide, thick slices.
Line a rimmed baking sheet with parchment paper, arrange the zucchini on it, and bake for 5 minutes.
Put the onion and garlic in a medium dry saucepan over medium heat and cook, stirring constantly, for 5 minutes. Add the spinach and cook, stirring occasionally, until wilted, 2 to 3 minutes. Let cool. Use your hands to squeeze all of the liquid out of the spinach mixture.
Drain the cashews and put them in a food processor. Add the tofu and granulated onion and process until smooth.
Layer the lasagne in a 13 x 9-inch baking pan in the following order: half of the noodles, 2 cups of the sauce, all of the mushrooms, all of the zucchini, all of the spinach mixture, another 2 cups of the sauce, the remaining noodles, another 2 cups of the sauce, all of the eggplant, and the remaining 2 cups of sauce. Cover with parchment paper, then with foil, and bake for about 1 hour, until the noodles are tender. Uncover, top with a single layer of the cashew mixture, and bake uncovered for 8 to 10 minutes, until the cashew mixture browns slightly. Serve hot.