Adjust the oven rack to the middle position and preheat to 475°F.
In a bowl, mix goat cheese, 1 tablespoon olive oil, thyme, garlic, lemon zest, ¼ teaspoon salt, and ¼ teaspoon pepper until combined.
Pat the chicken dry with paper towels. Using your fingers, carefully loosen the skin from the chicken to create a pocket. Spread about 1½ tablespoons of the cheese mixture under the skin of each chicken breast. Gently press on the skin to spread out the filling evenly. Brush the chicken skin with 1 tablespoon olive oil and sprinkle with ½ teaspoon salt and ½ teaspoon pepper.
Toss the sliced carrots with the remaining 1 tablespoon olive oil, brown sugar, ¼ teaspoon salt, and ¼ teaspoon pepper.
Arrange the chicken skin-side up on one side of a rimmed baking sheet.
Spread the seasoned carrots in a single layer on the other side of the sheet.
Roast until the chicken reaches an internal temperature of 160°F and the carrots are tender, 35–40 minutes. Rotate the sheet halfway through cooking.
Transfer the chicken to a serving platter and tent with aluminum foil. Let rest for 5 minutes. Toss the roasted carrots with the pan juices, season to taste with additional salt and pepper, and serve alongside the chicken.