In a large pot of boiling, salted water, cook the squash, covered, until tender, about 15 minutes. Drain the squash in a colander and let cool for 15 minutes.
Meanwhile, position a rack in the center of the oven and preheat to 400 degrees . In a small bowl, stir together the breadcrumbs and 2 tbsp. parmesan. Generously grease a shallow 2-qt. baking dish and sprinkle in the breadcrumbs.
Using a food processor, puree the squash. In a medium saucepan, melt the butter over medium-low heat; whisk in the flour and cook, stirring, for 2 minutes. Whisk in half of the milk, adding the remaining milk as the mixture thickens; continue to whisk for 1 minute, then add in half of the squash puree. When the mixture is heated through, remove from the heat. Whisk the egg yolks into the remaining puree, then transfer the mixture to the saucepan. Stir in the remaining 1 1/4 cups parmesan, the sage, 1/2 tsp. salt, 1/4 tsp. pepper and the nutmeg.
Using an electric mixer, beat the egg whites with a pinch of salt at medium speed just until stiff peaks form. Fold the whites gently into the squash mixture; scrape into the prepared baking dish. Bake until puffed and golden, about 30 minutes. Serve immediately.