Set a large soup pot or Dutch oven over medium-high heat and add the olive oil.

Add the chopped onion and sliced celery to the pot with a pinch of the salt. Cook for 4 to 5 minutes, stirring a few times, until the onion starts to soften.

Add the sliced mushrooms to the pot. Cook for 6 to 8 minutes, stirring now and then, until the mushrooms give off their liquid and then begin to brown around the edges.

Add the sliced garlic and chopped rosemary to the pot. Stir for 30 seconds, just until both smell fragrant.

Add the rinsed farro to the pot. Stir for 1 minute so the grains are coated with the oil and mixed through the vegetables.

Pour the chicken broth into the pot and stir, scraping the bottom of the pot with a wooden spoon to loosen any browned bits. Bring the broth to a boil over high heat.

When the broth boils, lower the heat to medium-low so the soup bubbles gently. Cover the pot partway with a lid and cook for 20 to 25 minutes, until the farro is tender but still pleasantly chewy.

While the soup cooks, pull the meat from the rotisserie chicken if needed and shred it into bite-size pieces. Set the chicken on a plate near the stove.

Check the farro in the pot. If it is still too firm in the center, cook the soup for 3 to 5 minutes longer.

Add the shredded chicken to the pot. Stir and cook for 3 minutes, until the chicken is heated through.

Stir in the lemon juice, half of the grated Parmesan, the remaining salt, and the black pepper. Taste the broth and add a little more salt if it needs it.

Let the soup sit off the heat for 2 minutes so the farro settles and the flavors come together.