Preparation

• Heat the oil over medium-high heat in a large (at least 8-quart), heavy-duty pot or Dutch oven. Add the onion, celery, and 1/2 Tbs. salt and cook, stirring occasionally, until the vegetables are softened and lightly browned in places, 4 to 6 minutes. Add the thyme and cook, stirring, until fragrant, 30 seconds.
• Add the broth and bring to a boil. Reduce to a simmer and cook, stirring, until the vegetables are completely tender, 20 to 30 minutes. 
• Stir in the reserved shredded chicken, barley, chard, and mushrooms. Simmer, stirring occasionally, until all are tender and the flavors meld, 5 to 10 minutes.
• Stir in 1 tsp. of the vinegar. Taste, and add more vinegar, 1 tsp. at a time, if needed, up to 2 Tbs. Season with salt and pepper to taste and garnish each bowl with a 1 Tbs. of the grated cheese, if you like.