Rehydrate the apricots by covering them with boiling water and steeping them for 10 minutes before draining and slicing into quarters. Pit and roughly chop the dates, mix them with the apricots, and set aside. Combine the sugar with all of the liquid ingredients. Bring to a boil and simmer 10 minutes, then set aside.
Season the Cornish Hens, inside and out, with salt and pepper. Place 1 bay leaf and 1/2 teaspoon of dried thyme inside each bird and close the body cavity by tieing with twine or with a skewer or toothpicks so that the bird holds its shape.
Brush the hens with olive oil, place them breast side down in a roasting pan, and roast for 20 minutes. At the end of 20 minutes, pour off any excess fat from the pan turn the birds breast side up and roast 20 minutes more. At the end of this 20 minute roasting period, add the apricots, dates, and the liquid to the roasting pan, and braise the contents of the pan together for another 25 minutes. Check the oven every so often to see that the liquid has not all evaporated. If it has, add enough water to keep the bottom third of the hens moist.
At the end of the cooking time, allow the hens to rest 10 minutes before serving. Place one hen on each dish and pour some of the apricot and date sauce over the top.