In a large stock pot, or Dutch oven, heat the olive oil over medium-low heat. Add in the onion, carrots, and leeks and sauté veggies, stirring occasionally, until soft and onions are translucent (about 10 minutes).
Stir in the thyme, garlic, salt, and pepper. Continue cooking for about 3 minutes, stirring frequently.
Add the chicken breasts, cauliflower, bay leaves, and then the broth on top of the veggies. Ensure the chicken and cauliflower are submerged in the broth. Bring to a boil, and then cover, reduce heat, and simmer until chicken is cooked to an internal temperature of 165-degrees F (this should take 10-15 minutes depending on how large your chicken breasts are).
Remove cooked chicken and bay leaves from soup (discard of the bay leaves). Using 2 forks, shred the chicken and set aside.
Using an immersion blender, puree the soup. If you want it extra creamy, puree for longer. If you want a less creamy base, puree for just a few seconds. Alternatively, you can transfer some of the soup to a blender to puree, then add back to the pot.
After the soup is blended, add the shredded chicken back to the pot. Let simmer for 10-15 minutes before serving. Enjoy!