Preheat the oven to 375 degrees F (190 degrees C).

Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain and rinse with cold water. Toss with 1 tablespoon olive oil to prevent sticking and add flavor.

Meanwhile, mix 2 cups mozzarella, ricotta, 1/2 cup Parmesan, tofu, and eggs together in a bowl. Combine remaining mozzarella and Parmesan in a smaller bowl.

While the noodles are cooking, heat remaining tablespoons olive oil in a skillet over medium-high heat. Add carrots, zucchini, onion, bell pepper, and mushrooms; saute until tender, 5 to 7 minutes. Add pasta sauce and spinach; simmer until sauce is heated through and spinach is wilted, about 3 minutes.

Cover the bottom of a 2-quart rectangular baking dish with a little of the vegetable sauce and arrange 1/2 of the lasagna noodles, overlapping slightly, over top. Layer with 1/2 of the cheese-tofu mixture over noodles and 1/2 of the remaining sauce. Repeat layers with remaining noodles, cheese-tofu mixture, and sauce. Top with mozzarella-Parmesan mixture.

Bake, covered, in the preheated oven for 30 minutes. Uncover and bake until bubbly, about 20 minutes more. Let stand for 10 minutes before slicing into 8 large squares and serving.