Take some of the cauliflower into floret like the chef did in the video and poach them separately from the soup in chicken stock or milk. You will add those at the end when the soup has been pulverized with the immersion blender.
In a large soup kettle, heat the butter and when hot add the onion and cook until translucent. (This could take a few minutes). Add the garlic and when fragrant add thyme, celery, carrots and potatoes and cauliflower.
Add the stock* be sure to watch how much the chef added in the video, you want to make sure you barely cover all your vegetables.
Cook for about 90 minutes or until all vegetables are soft and cooked.
Using your immersion blender, process the soup until all the ingredients are nice and smooth. Add the cream and the cheese; adjust the seasoning with salt and pepper. Add the shredded carrots, parsley and the reserved florets.