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Rinse chicken and pat dry. With a pair of kitchen shears, cut down each side of the backbone to remove it. Open chicken like a book and lay it skin-side down on a cutting board. Use a sharp kitchen knife to cut through the breastbone so you have 2 halves. Season chicken generously with salt and pepper on both sides. In a small bowl, combine minced rosemary, garlic, and 2 tablespoons olive oil. Rub marinade on both sides of chicken to coat well. Place chicken in a large sealable plastic bag and refrigerate at least 1 hour and up to 6 hours.
Heat oven to 400 degrees with rack in center. Remove chicken from marinade and pat dry; season with salt and pepper. Heat 2 tablespoons olive oil in a large, (12-inch) heavy, ovenproof skillet (preferably cast iron) over medium high. Place chicken halves in pan, skin-side down, and let cook, undisturbed, until they are light golden brown, about 4 minutes. Flip chicken over and place entire skillet directly in oven. Cook until chicken is a deep golden brown and a thermometer registers 165 degrees when placed in the thickest part of the thigh, about 40 minutes.
Remove pan from oven and place chicken on a plate or cutting board to rest while you prepare the sauce. Pour off excess fat, leaving any browned bits or juices in pan. Carefully return pan to stove (it will be very hot) over medium-high. Add vermouth, scraping up any browned bits with a spoon. Add chicken broth and rosemary sprig and cook, stirring occasionally, until sauce is reduced by half (you should have 1/2 cup total.) Cut chicken into 6 to 8 serving pieces and serve with pan sauce.