Heat olive oil in a large saucepan over medium heat. Add mushrooms, zucchini, onion, and carrot; cook and stir until softened, 5 to 10 minutes.
Stir spaghetti sauce, oregano, and basil into the saucepan; bring to a boil. Reduce heat and simmer sauce, covered, until flavors combine, 20 to 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking pan with cooking spray.
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.
Spread a thin layer of sauce in the bottom of the baking pan. Layer 6 lasagna noodles, 1 cup cottage cheese, and 1 cup mozzarella cheese on top of the sauce. Cover mozzarella cheese with spinach. Repeat layers with remaining sauce, lasagna noodles, cottage cheese, and mozzarella cheese. Sprinkle Parmesan cheese on top.
Bake in the preheated oven until bubbly, 40 to 45 minutes.