1. Preheat the oven to 400º. Cut the squash in half lengthwise, and scoop out and discard the seeds. Transfer the squash to a rack-lined baking sheet, cut-sides up. Place 1 tablespoon of butter and 2 sage leaves in the hollowed-out core of each half and season with salt and pepper.
2. Bake the squash until the flesh is tender, 45 minutes to 1 hour. Let the squash cool to room temperature. Using a fork, scoop out the meat of the squash from the skins into a small bowl. Reserve the skins for plating.
3. In a large sauté pan, melt the remaining 3 tablespoons of butter over medium heat. Add the walnuts and cook until they are toasted and the butter has started to bubble, 2 to 3 minutes. Add the remaining 11 sage leaves and toast until they become fragrant but not burnt, 1 minute. Mix in the roast squash and cook until fully warmed through, 2 to 3 minutes. Season with salt and pepper, and transfer to the reserved squash skins. Garnish with the Parmesan and serve.