Preheat the oven to 400 degrees F;. In a 10-by-14-inch baking dish, toss the carrots, parsnips, chopped onion and 3 sprigs rosemary with the chicken broth and 3 tablespoons water. Spread evenly in the pan.
Season the chicken cavity with salt and pepper, stuff with the halved onion and remaining 3 sprigs rosemary and arrange breast side down in the middle of the pan. Brush the top of the chicken with 1 tablespoon butter and season with salt. Roast the chicken and vegetables for 30 minutes. Turn the chicken breast side up, brush with another tablespoon of butter and season with salt and pepper. Roast until the juices run clear, about 1 hour longer. Transfer the chicken to a platter and cover loosely with foil.
Discard the rosemary sprigs from the vegetables in the pan. Using a slotted spoon, transfer the vegetables to a food processor, add the remaining 2 tablespoons butter and puree; season with salt and pepper. Reserve the pan.
Cut the chicken into serving pieces. Serve with the vegetable puree and any pan drippings.