Heat the oven: Preheat the oven to 425°F. Pat the chicken thighs very dry and season generously with salt and black pepper.
Sear the chicken: Heat a large oven-safe skillet over medium-high heat. Add the oil, then place the chicken in the skillet skin-side down. Cook for 10–12 minutes, shifting the thighs around the pan only as needed for even browning, until the skin is deeply golden and releases easily.
Add the vegetables: Flip the chicken. Add the parsnip, carrots, Brussels sprouts and thyme sprigs, tossing the vegetables in the rendered chicken fat. Drizzle with a little more oil if needed and season with salt and pepper. Arrange the vegetables in an even layer and place the chicken over the top, skin-side up.
Roast: Transfer the skillet to the oven and roast for 25–30 minutes, or until the vegetables are tender and the thickest part of each chicken thigh registers approximately 175°F without touching the bone.
Rest the chicken: Transfer the chicken and vegetables to a serving platter. Discard the roasted thyme stems. Carefully pour off the excess fat from the skillet, leaving approximately 1 tablespoon in the pan.
Sauté the aromatics: Return the skillet to the stovetop over medium heat. Add the shallot and cook for 4–5 minutes, stirring occasionally, until softened. Add the garlic and chopped thyme and cook for 1 minute.
Build the sauce: Sprinkle the flour over the shallot mixture and cook, stirring constantly, for 1–2 minutes. Slowly pour in the chicken broth and apple cider while whisking and scraping the browned bits from the bottom of the skillet.
Finish the sauce: Whisk in the Dijon mustard and sherry vinegar. Simmer for 5–10 minutes, stirring occasionally, until smooth and thick enough to lightly coat the back of a spoon. Taste and season with salt and pepper.
Serve: Spoon the sauce around the chicken and vegetables so the skin stays crisp. Serve the remaining sauce alongside.