2 medium-size butternut squash, peeled, seeded, and sliced into ½-inch slabs
salt and pepper to taste
4 medium onions, halved and thinly sliced
6 cloves garlic, crushed
3 sprigs fresh thyme
½ cup butter, divided
¼ cup flour
⅛ teaspoon nutmeg
3 cups milk
2 cups vegetable broth, divided
1–2 boxes dry lasagna noodles (the kind that should not be cooked in advance)
fresh sage, chopped
fresh basil, chopped
3 cups grated Parmesan
Preheat the oven to 425°F. Lay the squash one layer deep on an oiled baking sheet or in a shallow roasting pan. Season with salt and pepper and roast in batches until golden brown and fork-tender. Set aside to cool and reduce the oven temperature to 350°F.
While the squash is roasting, place the onions in a high-sided frying pan or shallow pot with a splash of olive oil and the garlic and sprigs of thyme. Place the pan over medium heat, cover, and cook until the onions are translucent and collapsing.
Remove the lid and continue cooking, stirring often, until they begin to caramelize. This can take up to 25 minutes. Season with salt and pepper to taste.
To make the béchamel, melt ¼ cup of the butter in a heavy bottomed saucepan over low heat, watching closely to make sure it doesn’t brown.
Make a roux by adding the flour and whisking to combine until smooth and uniform. Continue cooking, whisking continuously, until golden in color. Add the nutmeg.
Slowly pour in the milk and then 1 cup of the broth while whisking continuously over medium-low heat, until bubbling and thickened. Taste to ensure the raw flour taste has been cooked out. Season with salt and pepper. Allow to cool.
To assemble, use the remaining ¼ cup of the butter to heavily grease a 9-by-13-inch pan. Arrange a layer of the dry noodles in the pan and generously cover with some of the béchamel sauce, a layer of the squash, a smattering of the caramelized onions, a sprinkle of the fresh sage and basil, and a handful of the Parmesan.
Repeat the layering process until you reach the top of the pan, and then top with a layer of béchamel and Parmesan. Pour the remaining cup of broth carefully around the edges of the pan (this adds some moisture to help cook the noodles), being gentle to avoid disturbing the béchamel on top.
Place the lasagna on the middle rack of the oven and bake until the whole lasagna is bubbling (even in the center), about 45 to 60 minutes. Check for doneness by sticking a paring knife into the noodles; they should be tender.
If the top seems to be browning too quickly or drying out, cover the pan with tinfoil, removing it during the last 10 minutes so the top browns evenly.