Preheat your oven to 400°F (200°C). In a small bowl, mix dried thyme, paprika, salt, and pepper. Rub this spice blend evenly over the chicken pieces to season well.
Heat olive oil in a large oven-safe skillet over medium-high heat until shimmering. Place the chicken pieces skin-side down and sear for about 5 minutes until the skin is golden and crispy. Flip and cook for another 3 minutes. Remove the chicken from the skillet and set aside.
Add the minced garlic to the same skillet and cook for 30 seconds until fragrant, scraping up any browned bits from the bottom.
Arrange the cubed squash, halved Brussels sprouts, and sliced carrots evenly in the skillet around the chicken. Drizzle the vegetables with a little olive oil, and toss lightly to coat. Season with a pinch of salt and pepper.
Place the seared chicken back on top of the vegetables, skin-side up. Transfer the skillet into the preheated oven and roast for 35-45 minutes, or until the chicken reaches an internal temperature of 165°F and the vegetables are tender and caramelized.
Remove the skillet from the oven and let the chicken rest for 5 minutes to retain its juices. Serve hot, with the roasted vegetables spooned around or on top of the chicken.