Preheat the oven to 350 degrees F. Place the chicken in a roasting pan, tuck and fold the wings under the body and tie the legs together. Brush the chicken with the melted butter; season with salt and pepper. Roast the chicken for 60 minutes and then baste with the pan juices. Surround the bird with the boiled potatoes. Roast for 45 to 50 minutes more, or until the temperature in the deepest part of the chicken thigh registers 170 degrees F on an instant read meat thermometer. When the chicken is cooked, place the carrots in a pot, cover with a generous amount of water and boil until firm/tender. While that occurs, transfer the chicken to a large plate, tent with foil, and let rest 10 minutes. Place the potatoes in heatproof dish and keep warm in a 200 degrees F oven. Set the roasting pan on the stovetop, add the stock/flour mixture and simmer until a thickened gravy forms; season with salt and pepper. When the carrots are just tender, add the broccoli to the pot and cook for 2 minutes more. Drain the vegetables well. Cut the chicken into portions and serve with the vegetables and gravy.
Note: To cook the potatoes firm/tender, cover with cold water and boil until they are tender on the outside, but still a little firm in the middle. They will finish cooking along with the chicken when they are added to the roasting pan.