Combine the milk, squash and cornmeal in a big bowl. Cover with a towel and allow to soak. This can actually be done ahead, as early as the night prior, but a ten minute soak is sufficient, too.
Turn the oven on to 350.
Place a skillet over medium-high heat and add the onions with a generous splash of water. As soon as the water has cooked off, swirl in half of the butter and reduce the heat. Let the onions cook until they are just translucent. Whisk the cooked onions into the cornmeal squash mixture.
Generously grease the bottom & sides of a tall baking dish or soufflé pan with the remaining butter.
Separate the eggs. Whisk the yolks into the soaking cornmeal with half of the salt and the jarlsberg cheese, smoked paprika (optional) and nutmeg.
In a separate, warm to room temperature bowl, beat the egg whites & the remaining salt with an electric mixer or stand mixer until the egg whites are just becoming stiff.
Pour the whipped egg whites into the large bowl and using a spatula scraper, gently fold them together.
Pour the batter into the greased baking dish and bake for 35-45 minutes until a small spot in the center barely jiggles when shook. If there is a lot of movement throughout the dish, it needs probably needs another 10 minutes.
Serve as soon as possible, especially if it is puffed up. Dip a large spoon in and watch it settle and fall onto itself. It will still be a bit wet and hopefully full of good fluffiness.