Preheat your oven to 400°F (200°C). Line a large baking sheet with parchment paper for easy cleanup.
Pat the chicken breasts dry with paper towels, then season them generously with salt, pepper, and dried thyme on both sides.
In a small bowl, mix two tablespoons of olive oil with minced garlic and a pinch of salt, then brush this mixture over the chicken breasts for added flavor.
Arrange the seasoned chicken breasts on one side of the prepared baking sheet.
Toss the butternut squash cubes, sliced carrots, and halved brussels sprouts with the remaining one tablespoon of olive oil, ensuring they are evenly coated.
Spread the coated vegetables in a single layer on the opposite side of the baking sheet from the chicken, making sure they have space to roast evenly.
Place the baking sheet in the oven and roast for about 35 minutes, until the chicken reaches an internal temperature of 165°F and the vegetables are caramelized along the edges.
Halfway through roasting, use tongs to toss the vegetables gently for even caramelization, and baste the chicken with its own juices for extra tenderness.
Remove the sheet from the oven once everything is cooked through and golden brown. Rest the chicken for a few minutes before slicing.
Serve the sliced chicken alongside the roasted vegetables, spooning any pan juices over the top for added flavor.