Adjust oven rack to the upper-middle position and preheat to 475°F. In a large bowl, toss Brussels sprouts, shallots, and garlic with 2 teaspoons thyme, oil, 1 teaspoon rosemary, sugar, ¾ teaspoon salt, and ¼ teaspoon pepper. Spread in a single layer on a rimmed baking sheet, placing Season and arrange the chicken:
Pat chicken pieces dry with paper towels. Season with remaining ½ teaspoon salt and ½ teaspoon pepper. Place chicken skin-side up on top of the vegetables, arranging breasts in the center and thighs/drumsticks around the perimeter.
Combine melted butter, remaining 2 teaspoons thyme, 1 teaspoon rosemary, ¼ teaspoon salt, and ¼ teaspoon pepper. Brush the mixture over the chicken pieces. Roast in the oven until chicken breasts register 160°F and thighs/drumsticks reach 175°F, about 35 to 40 minutes. Rotate the sheet halfway through for even cooking.
Transfer chicken to a serving platter and let rest for 5 minutes. Toss vegetables with pan juices, then serve alongside the chicken.