Squash: Adjust an oven rack to the upper middle position and heat the oven to 450 degrees. Line a rimmed baking sheet with parchment paper. Toss the squash with 2 teaspoons of the oil, 1/4 teaspoon salt, and 1/8 teaspoon pepper and spread out over the prepared baking sheet. Roast the squash until tender and golden brown, about 30 minutes; set aside until needed.

Barley: Meanwhile, bring the broth and water to a simmer in a medium saucepan. Reduce the heat to the lowest possible setting and cover to keep warm.

Combine the remaining teaspoon oil and onion in a large saucepan. Cover and cook over medium-low heat, stirring occasionally, until the onion is softened, 8 to 10 minutes. Stir in the garlic and cook until fragrant, about 30 seconds.

Stir in the barley, increase the heat to medium, and cook, stirring often, until lightly toasted and aromatic, about 4 minutes. Stir in the wine and continue to cook, stirring often, until the wine has been completely absorbed, about 2 minutes. Stir in 3 cups of the warm broth and half of the roasted squash. Simmer, stirring occasionally, until the liquid is absorbed and the bottom of the pan is dry, 22 to 25 minutes. Stir in 2 more cups of the warm broth and continue to simmer, stirring occasionally, until the liquid is absorbed and the bottom of the pan is dry, 15 to 18 ­minutes longer.

Continue to cook the risotto, stirring often and adding a half cup of the remaining broth at a time as needed to keep the pan bottom from becoming dry (about every 4 minutes), until the grains of barley are cooked through but still somewhat firm in the center, 15 to 20 minutes longer. Off the heat, stir in the remaining roasted squash, Parmesan, butter, sage, and nutmeg. Season with salt and pepper to taste and serve.