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.