This is a delicious alternative to the classic meat-and-tomato-sauce lasagne. The squash comes on surprisingly big; roasting it caramelizes it and concentrates its flavor. I made a relatively light “cream sauce”—half milk/half chicken broth—because the two cheeses contribute a ton of richness.
The big news here is that you don’t need to precook the lasagne noodles. Cooking lasagne on the show one night, I was demonstrating a relatively new product called “no-boil” noodles, which are somehow processed so that you don’t need to precook them. A viewer called in and volunteered that she never needed to cook the noodles for her lasagne so long as she used enough sauce. I was skeptical, but she was right. We tried it for this dish, and it works just fine.
1 1/4 cups freshly grated Parmigiano-Reggiano
Preheat the oven to 450ºF. Toss the squash with the oil in a large bowl. Season with salt and pepper and spread out in one flat layer over the bottom of a large shallow roasting pan. Roast, stirring gently or turning once or twice, until almost tender, 15 to 20 minutes.
Meanwhile, combine the milk, stock, rosemary, and sage in a large saucepan. Bring to a boil over high heat. Remove from the heat, cover, and let steep for 10 minutes. Strain, discarding the herbs. Melt the butter in a separate saucepan over medium-high heat. Add the garlic and reduce the heat to medium-low. Cook, stirring often, until the garlic is tender and fragrant, 2 to 3 minutes. Add the flour and cook, stirring, for 6 minutes. Increase the heat to high, whisk in the strained milk, and bring to a boil. Reduce the heat to medium and simmer for 5 minutes. Season with salt and pepper. Cool slightly.
Reduce the oven temperature to 375ºF and butter a 13 x 9 x 2-inch baking dish. Spread a quarter of the sauce over the bottom of the dish. Cover with 3 lasagne sheets, arranging them so that they do not touch. Arrange half the squash slices on top. Spread a third of the remaining sauce over the pasta and sprinkle on half the mozzarella and 1/2 cup of the Parmigiano-Reggiano. Make one more layer in the same manner. Top with the remaining sauce and sprinkle on the remaining 1/4 cup Parmigiano-Reggiano. Cover the dish tightly with foil, and bake in the middle of the oven for 30 minutes. Remove the foil and bake for 15 minutes longer or until bubbling. Preheat the broiler to high and set the oven rack about 6 inches from the source of heat. Broil, watching carefully, and turning to distribute the heat evenly, until golden, 2 to 3 minutes. Let stand for 5 minutes before serving.