Canard aux Olives (Roast Duck with Olives)

Put fennel, parsley, peppercorns, bay leaf, and thyme on a 6" square piece of cheesecloth. Bring up corners of cheesecloth and tie with kitchen twine to make a bouquet garni; set aside. Heat 1 tbsp. butter and oil in an 8-qt. pot over medium-high heat. Add chicken pieces and reserved duck neck and gizzards, season with salt and pepper, and cook, stirring occasionally, until browned, 8 minutes. Add onions and cook, stirring occasionally, until soft, 5 minutes. Sprinkle with flour and cook, stirring, for 1 minute. Add reserved bouquet garni, stock, and wine and stir in tomato paste. Bring to a boil, reduce heat to medium-low, and simmer, occasionally skimming fat from surface, for 2 hours.

Meanwhile, bring a small saucepan of water to a boil, add olives, and cook for 2 minutes. Drain and rinse olives under cold water; set aside. Set a fine mesh strainer over a 4-qt. saucepan. Strain sauce, discarding solids. Add olives and heat over medium heat; cook, stirring occasionally, until sauce is reduced and coats the back of a spoon, 1–1 1⁄2 hours. Season with salt and pepper, cover, and keep warm.

Heat oven to 475°. Score duck skin all over with a small paring knife and rub skin with remaining butter; season with salt and pepper. Place duck breast side down on a rack in a large roasting pan and place in oven; roast for 20 minutes. Turn duck breast side up; roast for 20 minutes. Reduce oven temperature to 350°. Continue cooking until duck is browned and an instant-read thermometer inserted into the deepest part of a thigh (without touching the bone) reads 160°, about 1 hour. Transfer duck to a platter and let rest for 20 minutes.

To serve, carve duck into 8 pieces and arrange on a large platter to resemble the whole duck. Pour olive sauce over top and serve immediately.