Preparation

• Preheat oven to 350°F. If using whole chickens, cut chickens into skin-on, bone-in thighs, breasts, wings, and legs; discard back bone. Pat skin dry using paper towels. Sprinkle chicken all over with 2 teaspoons salt; set aside.
• Heat oil in a large Dutch oven over medium-high. Working in 3 batches, cook chicken, skin side down, until golden brown on all sides but not cooked through, 6 to 8 minutes per batch. Transfer chicken to a plate, and set aside.
• Reduce heat in Dutch oven to medium; add lardons, and cook, scraping up any brown bits from bottom of Dutch oven and stirring often, until lardons are browned, about 3 minutes. Add onions and garlic; cook, stirring constantly, until translucent, 2 to 3 minutes. Sprinkle sugar over top; add ½ cup butter, and stir well until butter is melted, about 2 minutes. Add celery and carrots; cook, stirring occasionally, until just starting to soften, about 3 minutes. Add mushrooms and dried thyme; cook, stirring gently, until softened, about 3 minutes. Add vermouth, scraping up any additional brown bits from bottom of Dutch oven. Stir in wine, chicken stock, rosemary sprigs, thyme sprigs, bay leaves, star anise, and remaining 1 teaspoon salt. Nestle chicken pieces into mixture in Dutch oven; bring to a simmer over medium.
• Stir together flour and remaining 1 tablespoon butter in a small bowl using a fork until combined. Add flour-butter mixture to Dutch oven, and gently stir to combine. Bake, uncovered, until meat is very tender and sauce is reduced and slightly thickened, about 2 hours, turning chicken a few times in sauce. Remove Dutch oven from oven, and let chicken rest for 30 minutes. Garnish with parsley, and serve.