Shred onion, apple and carrots either with a box grater or with shredding blade in food processor. Heat butter in large pot over medium heat. Add shredded onion, apple and carrot. Cook 7 minutes, until softened.
Stir in squash, broth, sugar, ginger, salt, cinnamon and cayenne. Bring to a boil over high heat. Reduce heat to medium-low and simmer 5 minutes.
Transfer mixture to food processor fitted with chopping blade or to blender. Puree until smooth (do this in batches, if necessary). Return to pot and stir in heavy cream. Gently heat through.