1 (4-pound) whole chicken, giblets and neck reserved
1 ½ teaspoons vegetable oil, divided
1 ½ teaspoons kosher salt, divided
½ teaspoon freshly cracked pepper
4 cups chicken broth, divided
½ medium onion, finely chopped
1 carrot, finely chopped
1 celery rib, finely chopped fine
4 sprigs fresh parsley
2 sprigs fresh thyme
1 large garlic clove, crushed
¼ cup dry white wine or dry vermouth
2 tablespoons butter
2 ½ tablespoons all-purpose flour
Adjust oven rack 12 to 13 inches from broiler element. Leave the oven off for the time being.
Remove the neck and giblets (if any) and place in a medium saucepan.
Spatchcock chicken by cutting out the back bone. Cut the back bone into pieces and place into the saucepan. Turn the chicken breast side up. Using the heel of your hands, firmly press on the breastbone to crack it and flatten it. Trim excess fat and skin from the chicken and place in the saucepan, along the neck and giblets (if any) and backbone. Set aside.
Using tip of paring knife, poke holes through skin over entire surface of chicken, spacing them approximately ¾ inch apart (this allows the fat go get rendered while cooking, which helps the skin crisp up).
Rub ½ teaspoon oil over the chicken skin; 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 place in oven, positioning skillet as close to the center of oven as possible. The handle should be facing out and turned to the corner.
Turn on broiler to highest setting, 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 (If the skin begins to brown too quickly, place foil loosely on top).
Meanwhile, make the gravy: Add 1 cup broth to the saucepan containing the reserved giblets, neck, backbone, and trimmings. Set over high heat. Cook, 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 the 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 the fond from the bottom of the pan. 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 is reduced by half, about 20 more minutes.
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. Pour the accumulated chicken juices from the pan into a bowl and allow to sit for 5 minutes. Skim off the fat and pour the juices into gravy. Mix well.