1. Preheat oven to 325 degrees. Spray a 13-by-9-inch casserole dish with nonstick cooking spray; set aside.

2. In an 8- to 10-inch skillet, melt 1 tablespoon butter over medium-high heat. Add mushrooms; cook until mushrooms begin to release their liquid, 3 to 5 minutes. Set aside.

3. In a medium saucepan, combine remaining 3 tablespoons butter and garlic over medium heat; cook for 1 minute. Add flour; cook, stirring, for 1 to 2 minutes. While whisking constantly, slowly add 1 cup each of milk and cream; continue whisking until thickened. Add remaining milk and cream, cream cheese, nutmeg, and cayenne. Season with salt and pepper. Reduce heat to low, and continue to simmer, stirring frequently, for 5 minutes. Remove from heat.

4. Combine 1 cup mozzarella with 1 cup Parmesan cheese; add to sauce, stirring until cheeses are melted. Stir in reserved mushrooms and liquid.

5. Spread a thin layer of sauce on bottom of prepared casserole dish. Place 3 lasagna noodles in the dish; sprinkle half the vegetables on top. Ladle half the remaining sauce over vegetables along with 1 cup mozzarella. Add 3 more lasagna noodles; repeat layers, ending with sauce. Top with paprika and remaining 1/2 cup each of mozzarella and Parmesan cheese; cover with foil. Bake until bubbling and golden brown, 50 to 55 minutes. Remove foil, increase heat to 350 degrees, and bake, uncovered, for 10 minutes more. Let stand for 15 minutes before serving.