Heat a tablespoon of olive oil in a large heavy-bottomed pot over medium heat until it shimmers and smells faintly nutty.
Add the diced onions and carrots to the pot. Cook, stirring occasionally, until the vegetables start to soften and release a sweet aroma, about 5 minutes.
Pour in the chicken broth and add the shredded cooked chicken along with the thyme sprigs and bay leaves. Bring the mixture to a gentle simmer.
Lower the heat and let the soup simmer uncovered for about 20 minutes, allowing the flavors to meld and the vegetables to become tender.
Remove the thyme sprigs and bay leaves. Taste the broth and adjust seasoning with salt and pepper as needed.
Squeeze fresh lemon juice into the soup, stirring to brighten the flavors and add a zesty punch.
Give the soup a final stir, taste for seasoning, and serve hot, garnished with fresh herbs or a slice of lemon if desired.