Dice the onion and celery. Cut the carrot into half coins and mince the garlic. Wash the cauliflower, then cut it into florets. Add the florets to a food processor and blitz until fine in texture, similar to rice grains.
In a pot over medium heat, add the onion, celery, garlic, and carrot. Sauté for about 5 minutes until soft and fragrant.
Transfer the sautéed aromatics to a plate. Add the chicken breast. Season with salt and black pepper. Sear the chicken for about 4 minutes on both sides until lightly brown on both sides.
Pour in the chicken broth. Add the bay leaf and fresh thyme. Bring the stock to a boil, then reduce it to a simmer. Cover and allow the chicken to fully cook through.
Remove the chicken from the pot and shred it using two forks, and return it back to the pot. Discard the bay leaf and add the cauliflower rice. Simmer uncovered for 5 minutes until the cauliflower is cooked.
Transfer the soup to a blender. Add most of the heavy cream, reserving a couple of tablespoons for garnish. Blend until smooth and creamy.
Taste and adjust the seasoning to your liking. Serve in bowls. Garnish with a drizzle of heavy cream, fresh parsley, and sunflower seeds.