How to Make It
Preheat oven to 450°F.
Line a baking sheet with parchment paper. Combine 1 tablespoon oil, 1/4 teaspoon salt, and potatoes in a bowl; toss to coat. Spread potato mixture on prepared pan; bake at 450°F for 15 minutes or until potatoes are tender when pierced with a fork. Remove pan from oven; keep warm.
Meanwhile, place a wire rack on a baking sheet. Heat 1 tablespoon oil in a large cast-iron skillet over medium-high. Sprinkle the chicken with 1/4 teaspoon salt and 1/4 teaspoon pepper. Add chicken to skillet, skin side down; cook 8 minutes. Flip, and cook 2 minutes (chicken will not be cooked through). Place chicken on wire rack (reserve drippings in skillet). Bake chicken at 450°F for 10 minutes or until done.
Return skillet to medium-high heat. Add remaining 1/4 teaspoon salt, remaining 1/4 teaspoon black pepper, halved Brussels sprouts, and carrots to drippings in pan; cook 8 minutes. Remove pan from heat. Remove vegetables from pan with a slotted spoon; set aside (do not wipe out pan).
Place shallots in a small bowl. Add vinegar to skillet, scraping to loosen browned bits. Add vinegar mixture, remaining oil, and brown sugar to shallots, stirring to combine. Place 1 cup potatoes and 1/2 cup vegetable mixture on each of 4 plates; top each serving with 1 chicken thigh and about 1 1/2 tablespoons shallot mixture.