Preheat the oven to 375°. On a large rimmed baking sheet, toss the diced squash with the whole shallots, sage and 1 tablespoon of the oil and season well with salt and pepper. Roast for 40 minutes, turning once, until the squash and shallots are softened and browned in spots. Crumble the sage. Leave the oven on.
In a large enameled cast-iron casserole, melt 1 tablespoon of the butter in the remaining 1 tablespoon of oil. Add the barley and cook over moderate heat, stirring, until golden, 7 minutes. Add the minced shallot and cook until softened. Add the Turkey Stock and bring to a boil. Season with salt and pepper. Cover and bake for 35 minutes, or until the barley is al dente.
Stir the chestnuts into the barley and simmer over moderate heat until the liquid is thickened and nearly absorbed, about 5 minutes. Stir in the remaining 2 tablespoons of butter, the roasted squash, shallots, sage and the parsley. Season with salt and pepper and serve.
The squash can be roasted one day ahead; refrigerate and return to room temperature before proceeding.