Heat oven to 350ºF with rack set in middle position.
Spread 1 ½ cups tomato sauce in bottom of a 9✕13-inch baking dish.
Arrange 4 lasagna sheets in an even layer over sauce (overlapping as needed).
Spread 1 ½ cups cottage cheese mixture over lasagna sheets.
Sprinkle sausage in an even layer over top.
Spread 1 ½ cups tomato sauce over meat.
Layer another 4 lasagna sheets over top.
Spread 1 cup cheese mixture over lasagna sheets.
Top zucchini slices, arranging in an even layer.
Spread 1 ½ cups tomato sauce over zucchini. Layer another 4 lasagna sheets over top.
Spread 1 cup cheese mix over lasagna sheets. Sprinkle spinach in an even layer over cheese mixture followed by cooked mushrooms.
Layer another 4 lasagna sheets over top. Spoon 1 ½ cups tomato sauce over top lasagna sheets. Spread remaining cheese mixture over top.
Sprinkle 1 ½ cups mozzarella and ½ cup grated Parmesan over top. Season with cracked black pepper.
Spray a large sheet of foil with nonstick spray (or coat in olive oil). Cover lasagna tightly with foil, oil-side down.