Step 1
Heat 4 Tbsp. butter in a large heavy pot over medium. Add fennel, onion, carrots, potato, thyme, and bay leaf; season with salt and pepper. Reduce heat to medium-low, cover, and cook, stirring occasionally and reducing heat if needed, until vegetables are soft but not browned and have released their moisture, 45–60 minutes (this will give them deep flavor).
Step 2
Add broth, bring to a boil, and season with salt and pepper. Reduce heat and simmer until potato is falling apart, 8–10 minutes. Let cool slightly. Remove herbs.
Step 3
Working in batches, purée in a blender until smooth. Strain into a clean pot; season with salt and pepper.
Step 4
Heat remaining 2 Tbsp. butter in a small saucepan over medium. Add chestnuts and cook, stirring often, until butter is browned and chestnuts are toasted, about 4 minutes.
Step 5
Mix crème fraîche and maple syrup in a small bowl. Serve soup topped with maple crème fraîche and chestnuts.
Step 6
DO AHEAD: Soup can be made 3 days ahead. Cover and chill.