Remove the hens from the refrigerator and let them sit for about 30 minutes to come to room temperature.
Combine the figs, honey, and water in a bowl and cover with plastic wrap. Let sit for 20 – 25 minutes to rehydrate the figs. Strain the figs and reserve the liquid for the glaze.
Preheat oven to 400 degrees.
Scatter the carrots and onion in the bottom of a roasting dish to create a rack for the hens to sit on. Add the rosemary springs.
Rinse and dry the Cornish hens. Rub the outside with olive oil and season the cavities and outsides of the birds with salt and pepper. Tie the legs together with kitchen twine and tuck the wings under the bird. Place in the roasting pan and roast for 30 minutes.
While the hens are roasting, prepare the glaze. Put the fig liquid and butter in a small saucepan and bring to a boil over medium-high heat. Reduce until syrupy, about 8-10 minutes. Add the glaze and let cook another minute. Let it cool, it will thicken more as it cools.
Baste the birds with the glaze and cook for another 5 minutes, baste again and cook for another 5-10 minutes or until an instant-read thermometer reaches 160 degrees when inserted into the thickest part of the thighs. Do one final glaze and let rest for 10 minutes before serving.
Strain the liquid from the bottom of the roasting dish into a small pitcher and serve with the hens.
The onions and carrots are delicious, you can serve as a side dish if desired.