1. Brine the chicken: Bring sugar, salt, vinegar, pepper, mustard seeds, garlic, thyme, bay leaf, and 12 cups water to a boil in an 8-qt. pot. Remove from heat; let cool for 10 minutes. Add ice and refrigerate brine until well chilled. Submerge the chicken pieces in the brine, cover, and refrigerate overnight.

2. Make the sauce: Heat 2 tbsp. oil in a 12" skillet over medium-high heat. Add the chicken wings in a single layer and cook, flipping once with tongs, until browned, about 8 minutes. Add the peppercorns, shallots, 2 sprigs thyme, and garlic and cook, stirring frequently, until shallots are soft, about 2 minutes. Add the madeira and cook until reduced by half, about 2 minutes. Add the broth, bring to a broil, and reduce heat to medium-low; simmer, stirring occasionally, until liquid has reduced to 1?4 cup, about 25 minutes. Skim fat from the sauce and strain sauce through a fine sieve into a 1-qt. saucepan; discard solids. Set sauce aside and cover to keep warm.

3. Meanwhile, heat oven to 425?. Remove chicken pieces from brine and pat dry with paper towels. Heat remaining oil in a 12" ovenproof skillet over medium-high heat. Add chicken skin side down and cook until golden brown, about 8 minutes. Add the remaining thyme sprigs and transfer the skillet to the oven. Cook until chicken is almost cooked through, about 15 minutes. Using tongs, flip chicken and cook until skin is crisp, 3 minutes. Transfer chicken to a plate and let rest for 5 minutes. To serve, spoon some of the sauce onto 4 plates. Halve thighs and cut breasts into 1" slices and fan out on plates. Spoon remaining sauce over top of chicken. Serve with sauteed carrots, if you like.