1. Heat the olive oil in a large pot over medium heat. Add the onion, carrots, and celery, then cook for 5 to 6 minutes until they start to soften.

2. Stir in the garlic and cook for 30 seconds until fragrant.

3. Add the cauliflower, chicken broth, salt, black pepper, and dried thyme. Stir well and bring the soup to a gentle boil.

4. Reduce the heat and let the soup simmer for 18 to 20 minutes, or until the cauliflower and carrots are very tender.

5. Lightly mash some of the cauliflower in the pot with a potato masher, or use an immersion blender for a few quick pulses to create a creamy texture while still leaving some chunks.

6. Stir in the shredded chicken, milk, and butter. Cook for another 5 minutes over low heat until everything is warmed through.

7. Taste and adjust seasoning if needed, then ladle into bowls and top with fresh parsley before serving.