Heat the large heavy-bottomed pot over medium heat. Add a splash of oil and sauté diced onions and minced garlic until fragrant and translucent, about 5 minutes, stirring occasionally.
Add the shredded chicken to the pot, stirring for 2-3 minutes until warmed through and lightly browned around the edges.
Stir in chopped carrots and celery, cooking for another 3-4 minutes until they start to soften and release their aroma.
Pour in the chicken broth, then add dried thyme, bay leaves, salt, and pepper. Bring the mixture to a gentle simmer over medium-high heat.
Reduce the heat to low and let the soup simmer uncovered for 20-25 minutes, until the vegetables are tender and flavors meld together.
Taste the soup and adjust seasoning with more salt, pepper, or herbs as desired. Remove the bay leaves.
Serve the soup hot, garnished with chopped fresh parsley for a bright finish.