Bring a medium pot with a couple of quarts of water to a boil. Add barley, bay leaf, onion and salt, and cook to tender, 25 minutes. Drain and reserve with a tiny drizzle of oil mixed in to keep it from sticking.
Place chicken in a pot with the poaching ingredients and enough water to just cover the chicken. Bring to a low, rolling boil, season water with salt and reduce heat. Poach 45-60 minutes, turning once; cool in poaching liquid to room temp.
Remove chicken from skin and bones, and dice or pull into bite-sized pieces. Strain stock. Use half the meat for the soup and reserve half for another meal or to add to a salad. You’ll have extra stock to refrigerate or freeze as well.
In a medium-size soup pot, heat oil, 4 turns of the pan, over medium to medium-high heat. Add mushrooms and brown. Add celery, garlic, shallots or onion, thyme, salt and pepper, and cook to soften. Add sherry or Marsala to deglaze and reduce by half. Stir in 2 quarts chicken poaching liquid and 2 1/2 to 3 cups of diced chicken. Wilt in the greens.
Just before serving, combine the barley with the soup and heat through.