Heat the oven to 450°. Coat the chicken quarters with 1 tablespoon of the oil, 1/2 teaspoon of the salt, and 1/8 teaspoon of the pepper. Arrange the chicken quarters, skin-side up, in a large roasting pan. Toss the cubes of butternut squash with the remaining 2 tablespoons of oil, 1/4 teaspoon of salt, and 1/8 teaspoon of pepper, and the sage. Add the cubes of squash to the roasting pan.
Cook, stirring the squash occasionally, until the chicken breasts are just done, about 20 minutes. Remove the pan from the oven and remove the breasts from the pan. Tilt the roasting pan and spoon off most of the fat from the pan. Return the pan to the oven. Continue cooking until the chicken legs and the squash are done, about 10 minutes longer. Remove the chicken and squash from the pan.
Pour off the fat from the roasting pan. Set the pan over moderate heat and add the water. Bring to a boil, scraping the bottom of the pan to dislodge any brown bits. Boil until reduced to approximately 2 tablespoons. Add any accumulated juices from the chicken. Spoon the sauce over the chicken.
• Instead of the butternut, use your favorite winter squash, such as acorn or Hubbard—or, to be really Italian, try pumpkin.
• Replace the chicken quarters with halved Cornish hens.