Put the fennel halves, cut-side up, in a baking dish and drizzle with enough olive oil to come up half-way; season with a generous amount of salt and pepper. Roast for 30 minutes or until completely tender - the roasted fennel is the base of the soup, so it's important the flesh is soft enough to be cut with a spoon. Reserve until ready to serve.
In a large pot, melt the butter over medium flame, add the onion and garlic, and saute for 4 minutes until translucent. Toss the thyme into the pan, along with the mushrooms, season with salt and pepper. Cook, stirring, for 10 to 15 minutes, until the mushrooms give off their liquid. Add the chestnuts and cook for a few more minutes until they have some nice color on them. Remove about 1 cup of the mixture to save as a garnish for the soup.
Pour the stock into the pot and bring to a boil. Reduce the heat to medium-low and simmer, uncovered, for 15 minutes, stirring now and then. Remove from the heat and puree the soup until completely smooth, using a handheld or regular blender. Stir in the cream to make the soup even richer, and simmer gently for 5 minutes. Taste for salt and pepper.
To serve: Ladle the soup into soup bowls; garnish each serving with a bit of the reserved mushrooms and chestnuts, a roasted fennel half, and a sprinkle of chopped chives.