Heat the oil in a large Dutch oven or roasting pan and add the turkey, skin side down. Cook over high heat until skin is golden, about 7 minutes. Remove the turkey and pour off the oil.
Place the pot back on the stove and add the water, onion, garlic, bay leaves, thyme, barley, salt and pepper. Replace the turkey, skin side up. Cover, reduce heat to low and simmer 1 hour.
Meanwhile slice the leeks into thin rounds. Heat the butter over low heat in a skillet, add the leeks, cover and cook 20 minutes, stirring occasionally. When the leeks are tender, remove from the heat and set aside.
Transfer the turkey to a cutting board. Carefully remove the meat from the bones and discard bones. Cut the breast meat against the grain into thin diagonal slices.
Remove the bay leaves from the barley mixture. Add the whipping cream and cook another 3 to 5 minutes or until barley mixture thickens. Pour the barley onto a large platter. Arrange the turkey on the barley and pile the leeks in the center of the platter.