Heat oven to 450° F with the rack in the highest position.
Toss the chicken, carrots, shallots, 1 tablespoon of the oil, and ½ teaspoon each salt and pepper on a rimmed baking sheet.
Roast, tossing the vegetables once, until the internal temperature of the thickest piece (avoiding the bone) registers 165° F, 25 to 30 minutes.
Meanwhile, combine the chives, vinegar, the remaining 2 tablespoons of oil, and ½ teaspoon salt in small bowl.
Serve the chive sauce over the chicken and vegetables.