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To brine the chicken, in a large bowl, whisk together the water, orange juice, salt, and cinnamon until combined. Place the chicken in a gallon-size resealable plastic bag and empty the brine into the bag with the chicken. Press out as much air as possible before sealing the bag and refrigerating it. Allow the chicken to rest in the brine overnight.
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Preheat the oven to 425 degrees Fahrenheit.
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For the glaze, in a small saucepan, bring the stock, blood orange juice, brown sugar, wine, duck fat, salt, cinnamon, allspice, and cardamom to a boil over medium-high heat, stirring occasionally. Reduce the heat to low and simmer until thickened slightly, about 10 minutes. Set aside.
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To roast the chicken, remove the chicken from the brine, rinse lightly, and pat dry. Rub it down inside and out with the duck fat, minced rosemary, and salt. Place the chicken in a roasting pan, place the rosemary sprig in the cavity, and truss. Arrange the blood orange slices in the pan around the chicken. Pour the glaze into the pan around the chicken and lightly brush the chicken with the glaze.
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Roast for 15 minutes. Reduce the temperature to 375[dg]F and continue roasting, brushing the bird with the pan drippings every 15 minutes, for 50 minutes to 1 hour more, until a thermometer inserted into the thigh of the bird reads at least 165[dg]F, tenting the pan with foil if the bird is browning too quickly.
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Allow the bird to rest for 20 minutes before carving and serving.