Broiled Chicken with Gravy

 Adjust oven rack 12 to 13 inches from broiler element (do not heat broiler). Place chicken breast side down on cutting board. Using kitchen shears, cut through bones on either side of backbone. Cut backbone into 1-inch pieces and reserve. Trim excess fat and skin from chicken and reserve with backbone. Flip chicken and use heel of your hand to press on breastbone to flatten. Using tip of paring knife, poke holes through skin over entire surface of chicken, spacing them approximately ¾ inch apart.

 Rub ½ teaspoon oil over skin and sprinkle with 1 teaspoon salt and pepper. Flip chicken and sprinkle bone side with remaining ½ teaspoon salt. Flip chicken skin side up, tie legs together with kitchen twine, and tuck wings under breasts. Heat remaining 1 teaspoon oil in broiler-safe 12-inch skillet over high heat until just smoking. Place chicken in skillet, skin side up, and transfer to oven, positioning skillet as close to center of oven as handle allows (turn handle so it points toward 1 of oven's front corners). Turn on broiler and broil chicken for 25 minutes. Rotate skillet by moving handle to opposite front corner of oven and continue to broil until skin is dark golden brown and thickest part of breast registers 155 degrees, 20 to 30 minutes longer. While chicken broils, make gravy.

 Bring 1 cup broth and reserved giblets, neck, backbone, and trimmings to simmer in large saucepan over high heat. Cook, adjusting heat to maintain vigorous simmer and stirring occasionally, until all liquid evaporates and trimmings begin to sizzle, about 12 minutes. Continue to cook, stirring frequently, until dark fond forms on bottom of saucepan, 2 to 4 minutes longer. Reduce heat to medium. Add onion, carrot, celery, parsley sprigs, thyme sprigs, and garlic to saucepan and cook, stirring frequently, until onion is translucent, 7 to 8 minutes. Stir in wine and bring to simmer, scraping up any browned bits. Add remaining 3 cups broth and bring to simmer over high heat. Adjust heat to maintain simmer and continue to cook, stirring occasionally, until stock (liquid only) is reduced by half, about 20 minutes longer.

 Strain stock through fine-mesh strainer set over bowl, pressing on solids to extract as much liquid as possible. Melt butter in now-empty saucepan over medium heat. Add flour and cook, stirring constantly, until mixture is deep golden brown, 5 to 8 minutes. Slowly whisk in stock. Increase heat to medium-high and bring to simmer. Simmer until thickened, about 5 minutes.

 Transfer chicken to carving board and let rest, uncovered, for 15 minutes. While chicken rests, transfer fat and drippings in skillet to small bowl and let sit for 5 minutes. Spoon off fat and discard. Whisk drippings into gravy. Season gravy with salt and pepper to taste. Carve chicken and serve, passing gravy separately.