Melt butter in a medium saucepan over medium-high heat. Add onions, celery, and 1 tsp. salt. Cook, stirring, until onion and celery have softened, about 5 minutes.
Add chestnuts, chicken stock, and bay leaves. Bring to a boil. Reduce heat to low, cover, and simmer until chestnuts are soft, about 10 minutes.
Remove bay leaves and purée soup using an immersion blender (or whirl in batches in a blender or food processor). Add more salt and freshly ground black pepper to taste. Serve garnished with a dollop of crème fraîche and a sprinkle of chopped parsley if you like.