Add the squash, apple, carrots and broth. Bring to boil. Reduce to a simmer, cover, and simmer for 30 minutes or so, until the squash and carrots are tender.
Use an immersion blender to puree the soup. If you don't have an immersion blender, you can use a standard blender and puree in batches and transfer back to pot.
Add pinches of nutmeg, cinnamon, cayenne, salt and pepper to taste. Stir in milk and cook over medium heat until warmed.