Remove the stem, cut the squash in half lengthwise, and scoop out the seeds. Place the squash cut side down on the prepared baking sheet and roast until fork tender, about 45 minutes. Scoop the cooked squash into a bowl and mash coarsely. You should have about 1-½ cups.
Meanwhile, heat 2 tablespoons of olive oil in the skillet. Add the leeks and sage and season with salt and pepper. Cover and cook over medium heat until golden, about 25 minutes.
In the saucepan, combine the water with the carrot, celery, and onion and bring to a boil. Reduce heat to medium and continue to cook for about 15 minutes. Strain out the vegetables. Whisk the polenta and kosher salt into the broth. Cook over very low heat, stirring frequently, until tender, about 35 minutes. Stir in 1 tablespoon of butter along with the mashed squash, pine nuts, leeks, Gouda, and half of the grated Grana Padano cheese.
Pour the hot polenta into the buttered cast-iron skillet and flatten the top of the polenta slightly without spreading it to the edges. Refrigerate until firm, about 3 hours.
Preheat the oven to 375° F/190° C. Rub the surface of the polenta with 1 tablespoon of softened butter and sprinkle with the remaining grated Grana Padano cheese. Bake in the preheated oven until the top and side are lightly browned and crisp, about 1 hour. Remove from the oven and allow to cool for 20 minutes. Cut into wedges.
While the polenta cools, heat 2 tablespoons of olive oil in the skillet. Sauté the mushrooms over medium heat until they begin to soften. Season with sea salt and pepper and continue to stir. Add the wine and continue cooking until the wine cooks down and the mushrooms are fork tender. Serve spooned over the polenta wedges.