Cook noodles in a large pot of boiling salted water until barely tender (8 minutes for dried, 1 minute for fresh). Drain and rinse under cold water. Spread the noodles on clean kitchen towels, cover with plastic wrap and set aside.

Combine sun-dried tomatoes and dried mushrooms in a small bowl. Add boiling water, cover and let stand for 10 minutes. Transfer the tomatoes and mushrooms to a cutting board with a slotted spoon and chop. Strain the soaking liquid through a fine sieve and set aside.

Heat 1 cup milk in a saucepan over medium heat until steaming. Meanwhile, put 3 tablespoons flour in a small bowl and gradually whisk in the remaining 1/3 cup milk until smooth; whisk into the hot milk and stir constantly over the heat until the sauce comes to a simmer and thickens. Continue cooking and stirring for 1 minute. Remove from the heat. Whisk in cream cheese, then 2/3 cup marinara sauce and vinegar. Season the sauce with salt and pepper; set aside.

Heat oil in a large nonstick skillet over medium-high heat. Add onion, carrot and garlic and saute until soft, about 2 minutes. Add fresh mushrooms, rosemary and the reserved tomatoes and porcini; cook until the fresh mushrooms are just wilted, about 2 minutes longer. Stir the remaining 1 teaspoon flour into the vegetables. Add the reserved soaking liquid and the remaining 1/3 cup marinara sauce and cook until the mixture thickens, about 1 minute. Remove from the heat and season with salt and pepper.

Preheat oven to 400 degrees F. Coat a 9-by-13-inch baking dish with cooking spray.

Smear the bottom of the prepared dish with 1/2 cup of the sauce. Line the bottom with a single layer of noodles. Spread half of the mushroom mixture over the noodles and sprinkle with 2 tablespoons of the Parmesan. Add another layer of noodles, arrange butternut squash on top and sprinkle with salt and pepper. Spread another 1/2 cup sauce over all. Add another layer of noodles, followed by the remaining mushroom mixture; sprinkle with 2 more tablespoons of the Parmesan. Finish with the remaining noodles and sauce. Sprinkle with the remaining Parmesan.

Coat a large piece of foil with cooking spray and tightly cover the baking dish. Bake the lasagna for 30 minutes. Uncover and bake until lightly browned and bubbling, 10 to 15 minutes more. Let stand for 10 minutes before serving.