Preheat the oven to 425°F (220°C). Lightly coat a 13x9x2-inch casserole dish with nonstick spray and spread 1 1/4 cups of the marinara sauce across the bottom.
Heat the olive oil in a large skillet over medium-low heat. Sauté the sliced onion until soft and beginning to brown, about 5 minutes.
Add the minced broccoli and minced mushrooms to the skillet with a pinch of salt. Cook until mushrooms release their water and the liquid evaporates, about 2–3 minutes.
Stir in the minced garlic and chopped kale and continue to sauté until the kale wilts, about 3 minutes. Remove from heat and let the vegetable mixture cool slightly.
In a large mixing bowl, combine the ricotta, 1 cup of the mozzarella, cottage cheese, egg white, chopped oregano, chopped basil, and black pepper until smooth.
Lay the cooked lasagna noodles flat on parchment or a clean work surface. Spoon about 1/4 cup of the cheese mixture onto each noodle and spread evenly, then top each with about 1/4 cup of the cooked vegetable mixture.
Starting at the short end nearest you, roll each noodle up tightly and place the rolls seam-side down in the prepared baking dish, arranging them so they are close but not touching.
Spoon 1 cup of the remaining marinara sauce evenly over the rolls. Sprinkle the remaining mozzarella and the grated Parmesan over the top.
Cover the dish with foil and bake for 20 minutes, until the cheese is hot and bubbly. Remove foil if desired to brown the top for 2–3 minutes before serving.