Preheat the oven to 425°F (220°C). Lightly coat a 13 x 9-inch casserole dish with nonstick spray and spread 1 1/4 cups of the marinara sauce on the bottom.
Heat the olive oil in a large skillet over medium-low heat. Add the sliced onion and sauté until soft and beginning to brown, about 5–7 minutes.
Add the minced broccoli florets, minced mushrooms, and a pinch of salt to the skillet. Cook until the mushroom liquid evaporates, about 2–3 minutes.
Add the minced garlic and chopped kale, then sauté until the kale is wilted, about 2–3 minutes. Remove from heat and let the vegetable mixture cool slightly.
In a large mixing bowl, combine the ricotta, 1 cup of the shredded mozzarella, cottage cheese, egg white, chopped oregano, chopped basil, and black pepper; stir until smooth and well mixed.
On a work surface lined with parchment, lay out the cooked lasagna noodles flat. Spread about 1/4 cup of the cheese mixture evenly over each noodle, then top each with about 1/4 cup of the cooked vegetable mixture.
Starting at the end closest to you, roll each noodle up tightly. Place the rolls seam-side down in the prepared casserole dish, arranging them so they are close but not touching.
Spoon 1 cup of the remaining marinara sauce evenly over the rolls, then sprinkle with the remaining 1/2 cup mozzarella and the grated Parmesan.
Cover the dish with aluminum foil and bake for 20 minutes, or until the cheese is hot and bubbly. Remove foil and serve with extra warmed marinara if desired.
