Preheat an oven to 450 degrees F (230 degrees C). Grease a baking sheet.
Brush the butternut squash halves with vegetable oil and season with salt. Roast in the preheated oven until golden and easily pierced with a knife, 45 to 50 minutes. Allow to cool for 15 to 20 minutes, then scoop the flesh into a bowl. Set aside.
Heat the butter in a large skillet over medium-low heat. Cook and stir garlic in the butter until softened. Stir in flour and cook for 3 minutes. Whisk in the milk until smooth. Bring to a simmer, and cook until thick, about 10 minutes, whisking occasionally. Stir in the butternut squash and season with salt and pepper. Sauce can be made 3 days ahead and refrigerated.
Reduce oven temperature to to 375 degrees F (190 degrees C). Grease a 9x13 inch baking dish.
Beat the heavy cream until foamy in a large glass or metal mixing bowl. Gradually add the salt, continuing to beat until medium peaks form. Lift your beater or whisk straight up: the tip of the peak formed by the cream should curl over slightly. Set aside.
Pour 1 cup of the butternut sauce into the baking dish and place 3 lasagna noodles on top in a single layer. Spread half of the remaining sauce over the noodles and sprinkle with 1/2 cup of Parmesan cheese. Place another layer of noodles and spread the remaining sauce on top and sprinkle with 1/2 cup of Parmesan cheese. Place the final layer of noodles on top. Spread the whipped cream over the final layer of noodles making sure the pasta is completely covered. Sprinkle with the remaining 1/3 cup of Parmesan. Cover baking dish tightly with aluminum foil.
Bake in the preheated oven for 30 minutes. Remove foil and continue baking until the top is bubbly and golden brown, about 10 minutes. Allow to rest for 5 minutes before serving.
