Preheat oven to 400 degrees F. In a large bowl, combine carrot, potato, parsnip or turnip, and onion. Add oil and balsamic vinegar; toss to lightly coat. Spread in a 9x9x2-inch baking pan; cover with foil. Roast for 30 minutes. Reduce oven temperature to 375 degrees F.
Meanwhile, gently separate the skin from the hen breast and tops of drumsticks by easing a paring knife or your fingers between the skin and the meat to make 2 pockets that extend all the way down to the neck cavity and over the drumsticks. In a small bowl, combine garlic, rosemary, salt, and pepper. Set aside 1 teaspoon of the fresh rosemary mixture (1/2 teaspoon if using dried rosemary). Rub remaining rosemary mixture under the skin onto the breast and drumsticks. Using 100%-cotton string, tie drumsticks to tail; tie wing tips to body. Sprinkle the reserved rosemary mixture on the skin. Uncover vegetables. Add hen to baking pan.
Roast hen and vegetables, uncovered, for 1 to 1-1/4 hours or until vegetables are tender and an instant-read thermometer inserted into the thigh of the hen registers 180 degrees F (the thermometer should not touch the bone), stirring vegetables once or twice. Remove string. Cover with foil; let stand for 10 minutes before serving.
If desired, garnish with fresh rosemary or sage leaves and tomatoes. To serve hen, use kitchen shears or a long heavy knife to carefully cut hen in half lengthwise. Remove skin and discard. Serve with vegetables. Makes 2 (1/2 of the game hen and 1 1/4 cup vegetables).