Heat a large heavy-bottomed pot over medium heat and add a tablespoon of olive oil. Once shimmering and fragrant, add the diced onions and carrots. Cook, stirring occasionally, until the vegetables soften slightly and smell sweet, about 5 minutes.
Add the shredded cooked chicken to the pot and stir to combine with the vegetables. Let everything warm through for a minute or two.
Pour in the chicken broth, making sure it covers the ingredients. Bring the mixture to a gentle simmer, listening for soft bubbling.
Add the thyme and bay leaves to the pot. Reduce the heat to low and let the soup simmer gently for about 20 minutes, allowing flavors to meld and vegetables to become tender.
Remove the bay leaves and taste the broth. Season with salt and pepper as needed, keeping the flavor bright and balanced.
Squeeze fresh lemon juice into the soup to add a zesty brightness, then give it a gentle stir.
Serve the soup hot, garnished with a sprig of thyme or a lemon wedge if desired, and enjoy its comforting, hearty texture.