Preheat the oven to 450°F.

Line a baking sheet with parchment paper. Combine 1 tablespoon oil, 1/4 teaspoon salt, and potatoes in a bowl and give it a toss to coat. Spread the resulting mixture on the prepped pan, and bake at 450°F for 15 minutes. You should be able to easily pierce the potatoes with a fork. Take the pan out of the oven and keep it warm.

Meanwhile, place a wire rack on a baking sheet. Heat 1 tablespoon of oil in a large cast-iron skillet on medium-high heat. Sprinkle the chicken with 1/4 teaspoon salt and 1/4 teaspoon pepper. Add the chicken to skillet, skin side down, and cook it for 8 minutes. Flip it over, and cook another 2 minutes (bear in mind, the chicken's not actually cooked through yet). Place the chicken on a wire rack and save the skillet drippings. Bake the chicken at 450°F for 10 minutes or until done.

Turn the skillet back up to medium-high heat. Add the remaining 1/4 teaspoon salt, remaining 1/4 teaspoon black pepper, halved Brussels sprouts, and carrots to the drippings in the pan, and cook all of that for 8 minutes. Remove the pan from the heat and remove the vegetables from it with a slotted spoon. Set the veggies aside, but don't wipe out the pan just yet.

Place shallots in a small bowl. Add vinegar to the skillet, and loosen any brown bits by scraping. Add the vinegar mixture, remaining oil, and brown sugar to the shallots, stirring to combine. Place 1 cup of prepped potatoes and a 1/2 cup of the veggie mixture on each of 4 plates.  Top each serving with 1 chicken thigh and about 1 1/2 tablespoons of the shallot mixture.