Heat a large heavy-bottomed pot over medium heat and add the olive oil, letting it warm until shimmering and fragrant.
Add chopped onions, carrots, and celery to the pot. Cook, stirring occasionally, until the vegetables soften and release a sweet aroma, about 8 minutes.
Toss in the shredded chicken to the pot, stirring to warm it through and allow the flavors to mingle with the roasted notes.
Pour in the chicken broth, then add the thyme and bay leaf. Bring the mixture to a gentle simmer, watching for tiny bubbling along the edges and a fragrant herby aroma.
Stir in the pasta or rice, and cook until tender, about 10 minutes, ensuring it remains firm to the bite.
Remove the pot from heat, then squeeze in fresh lemon juice and add lemon zest for brightness. Season with salt and pepper, tasting and adjusting as needed.
Let the soup rest off heat for 5 minutes to allow flavors to deepen and broth to slightly thicken.
Serve the soup hot, garnished with fresh herbs or a sprinkle of Parmesan, alongside crusty bread if desired.