Melt half of the butter in a large Dutch oven over medium-low heat. Add the shallot and the reserved squash seeds and fibers. Cook, stirring frequently, until the shallot is softened and the seeds are fragrant, about 4 minutes.

Stir in the water and a pinch of salt. Increase the heat to high and bring to a boil. Place the squash pieces on top of the liquid, skin-side down. Reduce the heat to low, cover, and simmer until the squash is completely tender when pierced with a paring knife, 30 to 40 minutes.

Transfer the cooked squash to a rimmed baking sheet. Strain the cooking liquid through a fine-mesh strainer into a large bowl, discarding the solids. Return the strained liquid to the pot.

Scrape the squash flesh from the skin into the pot with the liquid using a large spoon; discard the skins. This is much easier than peeling raw squash and extracts more flavor.

Puree the soup until completely smooth using an immersion blender, or process in batches in a standard blender until velvety.

Stir in the heavy cream, dark brown sugar, and nutmeg. Reheat over medium heat until hot but not boiling. Whisk in the remaining butter, and season with salt and pepper to taste.

Serve hot in warmed bowls.