Heat 1 tablespoon oil in a Dutch oven over medium-high heat; swirl to coat. Add onion and garlic; sauté for 3 minutes or until tender. Add mushrooms. Cook for 8 minutes or until liquid almost evaporates. Stir in oregano, 1/4 teaspoon salt, black pepper, nutmeg, and tomato sauce. Cook 2 minutes, stirring frequently. Remove from heat; set aside.
Combine eggs and parsley in a large bowl. Heat the remaining 1 teaspoon oil and butter in a medium saucepan over medium heat. Sprinkle flour evenly into pan; cook 2 minutes, stirring constantly. Gradually add milk to flour mixture, stirring with a whisk until smooth. Bring to a boil; cook 2 minutes or until thickened, stirring frequently. Remove from heat; let stand 4 minutes. Stir in remaining 1/4 teaspoon salt and 1 cup cheese. Gradually add hot milk mixture to egg mixture, stirring constantly with a whisk. Add pasta to milk mixture; toss to combine.
Spread 2 cups pasta mixture in an 11 x 7-inch baking dish coated with cooking spray. Top with mushroom mixture. Top with remaining pasta mixture. Cover with foil coated with cooking spray. Bake at 350° for 30 minutes. Remove foil; sprinkle with remaining 1/2 cup cheese.
Broil 5 minutes or until cheese melts. Let stand 15 minutes.
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