Heat oven to 425° F with the racks in the upper and lower thirds. Combine the butter, thyme, ¾ teaspoon salt, and ½ teaspoon pepper in a small bowl.
Slide your fingers under the skin of the breasts, thighs, and drumsticks to loosen the skin. Spread the butter mixture underneath the skin of these pieces and on the outside of the wings.
Divide the carrots and onions between 2 rimmed baking sheets. Add 2 cups water to each pan; top with ovenproof wire racks. Place the breasts on one of the prepared sheets and roast, on the top rack of oven, for 30 minutes.
Place the thighs, drumsticks, and wings on the second prepared sheet. Switch the breasts to the bottom rack of oven and place the high pan on the top rack. Continue to roast until an instant-read thermometer inserted in the thickest part of a breast and thigh registers 165° F, 30 to 35 minutes more. (Add 1 cup more water to each pan if the vegetables begin to scorch.)
Transfer the thighs, drumsticks, and wings to a serving platter. Transfer the breasts to a cutting board and let rest for 15 minutes before slicing. Serve warm. Reserve the vegetables and any pan juices for the gravy.