1. Preheat oven to 375°.
2. Heat a large skillet over medium-high heat. Add oil to pan; swirl to coat. Add onion; sauté 2 minutes. Add mushrooms; sauté 6 minutes. Add garlic; sauté 30 seconds. Transfer mushroom mixture to a bowl.
3. Return pan to heat; add half of spinach and 1/4 cup water. Cook 2 minutes or until spinach wilts, turning with tongs. Add cooked spinach to mushroom mixture. Repeat procedure with remaining half of spinach and remaining 1/4 cup water. Stir vegetable mixture until blended.
4. Combine eggs and next 3 ingredients (through pepper) in a bowl, stirring with a whisk. Stir in milk and ricotta cheese.
5. Pour 1 cup pasta sauce into bottom of a 13 x 9-inch glass or ceramic baking dish coated with cooking spray. Arrange 4 lasagna noodles over sauce. Top with one-third of ricotta mixture, one-third of vegetable mixture, and 1/3 cup mozzarella cheese. Repeat layers twice with sauce, noodles, ricotta mixture, vegetable mixture, and mozzarella cheese. Spread 1 cup sauce over mozzarella. Arrange 4 noodles over sauce. Top with 1 cup sauce, 1 cup mozzarella cheese, and Parmesan cheese.
6. Cover with a sheet of foil coated with cooking spray, coated side down. Bake at 375° for 45 minutes. Remove from oven, and let stand 15 minutes.