In a large saute pan, heat ¼ cup oil over medium heat, add the onions or shallots, turnip and chopped thyme. Cook over moderate heat until the turnips are browned and just tender, about 10 minutes. Add the chopped greens and cook until just wilted and tender, another 5-8 minutes. Add the cooked turnips and greens to the barley, pour on remaining dressing, toss well, season with additional salt and pepper if desired. Serve warm as a side dish or room temperature as a salad.