Cook lasagna noodles according to package directions, omitting salt and fat. Drain; set aside 9 noodles.
Heat oil in a nonstick skillet over medium heat. Add mushrooms; sauté 3 minutes. Add nutmeg and garlic; sauté 5 minutes. Set aside.
Combine ricotta and next 5 ingredients (ricotta through egg whites); set aside.
Preheat oven to 375°.
Spread 1/2 cup marinara sauce in bottom of a 13 x 9-inch baking dish coated with cooking spray. Arrange 3 lasagna noodles over sauce; top with half of ricotta cheese mixture, half of mushroom mixture, 1 1/2 cups sauce, and 1 cup mozzarella cheese. Repeat layers, ending with noodles. Spread 1/2 cup sauce over noodles.
Cover and bake at 375° for 40 minutes. Uncover; sprinkle with remaining 1 cup mozzarella cheese and Parmesan cheese; bake 10 minutes. Let stand 10 minutes before serving. Garnish with oregano leaves, if desired.