Preheat the oven to 375 F. Take one quarter of the cored cauliflower and cut it in to 1/2-inch thick flat, floret slices. Set aside.
Coarsely chop the remaining cauliflower.
In a Dutch oven or soup pot, heat the olive oil over medium heat. Add the onion and a pinch of salt and cook, stirring occasionally, until softened, about 3-4 minutes. Stir in the diced potato and the chopped cauliflower, then add the chicken stock and the thyme sprigs. Bring to a boil, then reduce the heat to medium low and simmer, stirring occasionally, until the vegetables are very tender, about 30 minutes. Discard the thyme sprigs.
While soup is simmering, place reserved cauliflower floret slices in an even layer on one end of a baking sheet. Prepare the Brussels sprouts as directed above and scatter over the other end of the baking sheet. Drizzle with a bit olive oil and season with salt. Roast for 5 minutes in the preheated oven. Remove from oven and transfer Brussels sprout leaves to a plate. Return to oven and roast a further 10-15 minutes, or until the cauliflower florets are golden on the bottom. Remove from oven and transfer florets to a plate, with the golden side up.
Using an immersion blender or in two batches in a blender, puree the soup until very smooth. Return the soup to the pot and stir in the cream. Warm over moderate heat. Taste soup and season generously with salt and some freshly ground pepper, to taste.
To serve, ladle into bowls and top with some roasted cauliflower florets and roasted brussels sprout leaves. Sprinkle with crumbled goat cheese and a grind of freshly ground pepper.
The soup can be made ahead and refrigerated for up to 3 days. Reheat gently before serving, then top with garnishes.
Be sure to read the "Cook's Notes" in the original post, for more tips, options, substitutions and variations for this recipe!