In a large soup pot over medium high heat, add 2 tablespoons olive oil. When the olive oil is shimmering, add the onions and garlic and sauté for about 5 minutes. Add the cauliflower and season with salt and pepper. Cook for 15 minutes or until cauliflower is tender and the onions are soft. Add the sherry vinegar and deglaze the pan. Add the vegetable stock and simmer 30 minutes. Using a handblender, puree until smooth.
Meanwhile, heat 2 tablespoons olive oil in a large sauté pan. Add the chestnuts, season with salt and pepper, and sauté a few minutes minutes. Add the vermouth, bay leaves, and thyme, and reduce halfway. Remove herbs and puree with a cup of water or more until smooth. (It can be a bit chunky and thick here, no worries.)
Stir the parmesan into the cauliflower puree, and then mix the two purees together. Add the cream and the pepper, a few big pinches of salt, and puree. Taste and adjust seasoning.
Serve with a sprig of thyme and a few grinds of black pepper on top as garnish.