Heat the oven to 425°F/220°C. Toss the squash cubes with olive oil, chopped sage, salt, and pepper on a sheet pan 33 x 46 cm / 13 x 18 inch, spreading them in a single layer.

Roast for 25 to 30 minutes, stirring once halfway, until the edges caramelize and a fork slides in easily. Drain any liquid on paper towels for 5 minutes.

While the squash roasts, melt the butter in a 3-quart / 2.8 L saucepan over medium heat. Add the garlic and whole sage leaves and cook for 2 minutes until fragrant.

Whisk in the flour and cook for 1 to 2 minutes, whisking constantly, until it smells toasted and looks pale gold.

Pour in the warm milk in a slow stream, whisking the whole time to keep it smooth. Simmer for 8 to 10 minutes, whisking often, until the sauce coats the back of a spoon. Stir in the nutmeg, salt, and pepper, then fish out and discard the sage leaves.

Reduce the oven to 375°F/190°C. Spread a thin layer of bechamel across the bottom of a 23 x 33 cm / 9 x 13 inch baking dish.

Layer 3 noodles over the sauce, then a third of the roasted squash, a third of the remaining bechamel, and a third of the mozzarella and parmesan. Repeat the layers twice more, finishing with noodles, the last of the bechamel, and the remaining mozzarella and parmesan on top.

Cover with foil and bake for 25 minutes. Uncover and bake for another 15 to 20 minutes, until the top is bubbling and golden brown and a thermometer at the center reads at least 165°F/74°C.

While the lasagna rests, melt the butter in a 20 cm / 8-inch skillet over medium heat. Fry the remaining sage leaves for 30 to 60 seconds per side, until crisp and the edges curl. Transfer to a paper towel.

Let the lasagna rest for 10 minutes before slicing. Top each slice with the crispy sage leaves and a drizzle of the brown butter.