This vegetarian butternut squash lasagna pairs thinly sliced squash with golden shallots, walnuts, fresh sage, garlic béchamel, mozzarella, and Parmigiano-Reggiano. The squash bakes directly in the layers, so there is no need to roast it first.
Prep the squash. Place each squash half flat-side down and cut it crosswise into 1/4-inch-thick slices. You should have about 8 cups, or 2 pounds, of sliced squash.
Sauté the aromatics. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the shallots and cook, stirring, until golden brown, about 3 minutes. Add the walnuts and sage and cook until the sage wilts, about 2 minutes. Season with 1/2 teaspoon kosher salt and a generous amount of black pepper.
Make the garlic béchamel. Heat 1/4 cup olive oil in a saucepan over medium-low heat. Add the garlic and cook for 1 minute. Whisk in the flour and cook, stirring, for 3 minutes. Increase the heat to medium and add the milk a few tablespoons at a time, whisking until smooth before adding more. Once all the milk is incorporated, continue cooking and whisking often until the sauce reaches a simmer, about 20 to 25 minutes. Reduce the heat to low and whisk until the sauce is thick enough to coat the back of a spoon.
Season the béchamel sauce: Whisk in 1/2 teaspoon kosher salt, several grinds of black pepper, and the nutmeg. Remove from the heat. The sauce should be thick but still pourable. Taste and add another pinch of salt if needed.
Combine the cheeses. In a small bowl, toss together the mozzarella and Parmigiano-Reggiano.
Assemble the lasagna. Preheat the oven to 375°F. Brush a 9-by-13-inch glass baking dish with olive oil. Spread about 3/4 cup of béchamel evenly over the bottom, then arrange 3 noodles on top. Spread another 3/4 cup of béchamel over the noodles. Add one-quarter of the squash slices, overlapping them slightly, and season with salt and pepper. Top with one-quarter of the shallot-walnut mixture and one-quarter of the cheese mixture. Repeat three more times, beginning with the noodles and ending with the cheese. Add a few fresh sage leaves on top, if you like.
Bake the lasagna. Cover the dish with foil and bake for 1 hour. Remove the foil and insert a sharp paring knife through the center. It should slide easily through the squash and noodles. If the squash is still firm, cover the dish again and bake for up to 15 minutes more.
Broil the top. Remove the foil and switch the oven to broil. Broil until the topping is bubbling and golden brown, about 5 to 10 minutes. Watch carefully to prevent burning.
Rest and serve. Let the lasagna cool for 10 minutes before slicing. Garnish with fried sage leaves, if you like.
Squash quantity: If your squash weighs more than 2 1/2 pounds, measure out 2 pounds, or about 8 cups, of sliced squash. Reserve the rest for another use.
To halve the recipe: Halve all the ingredients and assemble the lasagna in an 8-by-8-inch glass baking dish. Because this dish fits 2 noodles per layer, make three layers rather than four and divide the filling, sauce, and cheese into thirds.
Make ahead: Trim, peel, and slice the squash and refrigerate it in a zip-top bag overnight. You can also sauté the shallot, walnut, and sage mixture and refrigerate it overnight.
Make the béchamel ahead: Cool the sauce completely, then press plastic wrap directly against the surface to prevent a skin from forming. Refrigerate for up to three days. Reheat over low heat, whisking until smooth, before assembling the lasagna.
Assemble ahead: Assemble the lasagna, cover it, and refrigerate it before baking. Let it come to room temperature before transferring it to the oven.
Store and reheat: Refrigerate leftovers for up to four days. Cover individual servings lightly with foil and reheat in a 350°F oven for about 15 minutes.