Preparation

Roast the squash and garlic

• Position a rack in the center of the oven and heat the oven to 425°F.
• Put the squash cut side up on a large, heavy-duty rimmed baking sheet. Put 1 garlic clove and 1 sprig of thyme in each cavity. Drizzle each half with 1 tsp. of the oil and then season each with 1/4 tsp. salt and a few grinds of pepper. Roast until the squash is browned in spots and very tender when pierced with a skewer, 45 to 50 minutes. Remove from the oven and let cool completely. 
• Discard the thyme sprigs. Peel the garlic and put in a large bowl. Scoop the squash flesh from the skins and add it to the garlic. Mash with a fork until smooth. Season to taste with salt and pepper.
• Reduce the oven temperature to 350°F.

Make a goat cheese and sage sauce

• Melt 4 Tbs. of the butter in a 3-quart saucepan over medium heat. Add the sage sprigs and cook until the butter is fragrant, 2 to 3 minutes. Discard the sage. Add the flour and whisk until smooth and golden, about 2 minutes. Gradually whisk in the milk and cook, whisking occasionally, until thickened enough to coat the back of a spoon, about 15 minutes. Stir in the goat cheese, 1/2 cup of the pecorino, the nutmeg, 1 tsp. salt, and a few grinds of pepper. Season to taste with more salt and pepper. Set aside 1-1/2 cups of the sauce and mix the rest with the mashed squash. 

Season the breadcrumbs

• Melt the remaining 1 Tbs. butter in a small skillet over medium heat. In a small bowl, mix the breadcrumbs, chopped sage, chopped thyme, a pinch of salt, and few grinds of pepper. Add the melted butter and mix well. 

Assemble and bake the lasagne

• Spread 1/2 cup of the reserved cheese sauce over the bottom of a 9x13x3-inch baking dish. Cover the sauce with a slightly overlapping layer of cooked noodles, cutting them as needed to fill any gaps. Spread 1 cup of the squash mixture evenly over the noodles. Add another layer of noodles and repeat the layers as instructed above, to make a total of 4 squash layers and 5 pasta layers. Spread the remaining 1 cup cheese sauce evenly over the top. Sprinkle with the breadcrumb mixture and the remaining 1/2 cup pecorino.
• Cover the baking dish with foil and bake for 40 minutes. Remove the foil and bake until the top is browned and bubbly, 15 to 20 minutes. Cool for at least 10 minutes before serving.

Make Ahead Tips

You can make the roasted squash mixture up to 1 day ahead of assembling the lasagne.

You can assemble the lasagne up to 2 days ahead of baking it. Tightly wrap the baking dish in plastic and refrigerate it. Let the lasagne come to room temperature before baking it.