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.
In a large pot, melt the butter over medium flame; add the onion and garlic, 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, the roasted fennel, and a sprinkle of chopped chives.