Chop the carrots into thick slices, dice the onion, and chop the celery into small pieces for even cooking.
Heat your large pot over medium heat and add the olive oil. Once shimmering, add the diced onions and sauté until translucent, about 5 minutes, until they smell sweet and the edges are just starting to turn golden.
Add the sliced carrots and chopped celery to the pot. Cook, stirring occasionally, until the vegetables start to soften and release their aroma, about 8 minutes. They should be tender but still hold some bite.
Pour in the chicken broth and add the dried thyme and bay leaf. Bring the mixture to a gentle simmer, allowing the flavors to meld, about 10 minutes.
Stir in the shredded cooked chicken, and let everything simmer gently for another 15-20 minutes, stirring occasionally. The broth should become fragrant, and the vegetables tender.
Taste the soup and season with salt and pepper as needed. Remove the bay leaf and discard. Serve hot, garnished with chopped fresh parsley and enjoy the comforting, hearty flavors.