Heat oil in a nonstick skillet over medium-high heat until shimmering. Add the majority of the squash cubes in an even layer and cook without stirring until golden brown, about 5 minutes. Season with salt and pepper and continue to cook, stirring occasionally, until the squash is tender and browned, about 5 more minutes, then transfer to a bowl.

Add the remaining squash cubes and the reserved squash fibers and seeds to the skillet. Cook over medium heat, stirring often, until lightly browned, about 4 minutes. Transfer to a large saucepan and add the broth and water. Cover the pot and bring to a boil over high heat, then reduce the heat to medium-low and simmer gently for 10 minutes. Strain the broth through a fine-mesh strainer into a clean pot, pressing on the solids to extract as much liquid as possible, then discard the solids. Keep the strained broth warm over low heat.

Melt most of the butter in the skillet over medium heat. Add the onions, salt, and pepper and cook until the onions are softened, 5 to 7 minutes. Add the garlic and cook until fragrant. Stir in the rice and cook, stirring often, until the grain edges begin to turn translucent, about 3 minutes.

Stir in the wine and cook, stirring constantly, until fully absorbed, 2 to 3 minutes. Stir in the majority of the hot broth mixture and half of the browned squash cubes. Reduce the heat to medium-low, cover, and simmer until almost all the liquid has been absorbed and the rice is just al dente, 18 to 19 minutes, stirring twice during cooking.

Add the remaining hot broth mixture and stir gently and constantly until the risotto becomes creamy, about 3 minutes. Stir in the Parmesan, remove from the heat, and cover for 5 minutes to rest. Stir in the sage, nutmeg, and the remaining butter. Season with salt and pepper to taste and gently fold in the remaining browned squash cubes. Serve hot, adjusting the consistency with any leftover broth if needed.