Preheat the oven to 350ºF. Grease a 9×12-inch (or similar size) baking dish.

Heat 2 tablespoons of oil in a large skillet over medium heat. Add the onions and cook, stirring, for 10 minutes. Add 1 teaspoon salt and ½ teaspoon pepper. Continue to cook, stirring occasionally, until the onions are very soft and brown, about 35 minutes more. Add the balsamic vinegar and cook, stirring, until the vinegar evaporates. Remove from heat. Stir in the sage. Season to taste with more salt and pepper if desired.

While the onions are cooking, roast the squash and make the polenta: Place the squash on a rimmed baking sheet, drizzle with 2 tablespoons of olive oil, and season with 2 teaspoons salt and 1 teaspoon pepper. Roast the squash, stirring twice, until tender and starting to brown, 20-30 minutes. Mash the squash with a fork or potato masher. Leave the oven on.

Bring 6 cups of water and 2 teaspoons of salt to a boil in a large, heavy sauce-pan. Slowly stir in the polenta and continue to cook, stirring occasionally, until thickened, about 20 minutes. Stir in the butternut squash and Parmesan cheese.

Place half the polenta mixture in the bottom of the prepared baking dish and smooth into an even layer. Add half the onion mixture and half the shredded cheese. Repeat the layers. Bake for 15 minutes or until the cheese is melted.