Brown the Chicken: Place a large pot over medium-high heat. Add a tiny bit of butter and place the chicken thighs in, skin-side down. Cook for 5 minutes until the skin is golden and crispy. Flip and cook for 2 minutes. Remove the chicken from the pot and set it aside on a plate (it won't be cooked through yet).
Sauté the Veggies: In the same pot (using the chicken fat left behind), add the onion, celery, and mushrooms. Sauté for about 5 minutes until softened. Stir in the minced garlic for the last 30 seconds.
Simmer: Pour in the chicken broth and add the dried herbs. Place the whole chicken thighs back into the liquid. Bring to a gentle boil, then turn the heat down to low. Cover and simmer for 20 minutes.
Add Cauliflower: Add the cauliflower florets to the pot. Simmer for another 10 minutes until both the chicken and cauliflower are tender.
Shred the Meat: Use tongs to lift the chicken thighs out onto a cutting board. Use two forks to pull the meat off the bones. Discard the bones (and skin, if you prefer). Chop the meat into bite-sized pieces and stir it back into the soup.
Finish with Cream: Stir in the heavy cream. Let it warm through for 1–2 minutes, but do not let it boil. Season heavily with salt and pepper.