Melt the butter in a large, heavy pot (such as a Dutch oven) set over medium heat. When the butter has stopped foaming, add the celery root and onions and salt lightly. Cook until the celery root and onions are turned translucent (but have not browned), stirring often, about 15 minutes.
Add the apples and the broth. Cover and bring to a boil. Reduce the heat to medium-low and simmer, covered, until the celery root and apples are tender, stirring occasionally, about 30 minutes.
Working in batches, purée the soup in a blender until smooth, adding more broth as needed to achieve desired consistency. Return soup to a clean pot set over medium low heat. Gently re-warm the soup and stir in the heavy cream. Season to taste with salt and pepper.
Ladle soup into individual bowls. Sprinkle some of the chives on top of each bowl for garnish and serve.
Serves 6 to 8.