Saute the vegetables. In a frying pan over medium heat, melt the butter. Add the onion and saute until softened, about 5 minutes. Add the ginger, cinnamon, and nutmeg and saute until fragrant, about 1 minute longer.
Cook the soup. Put the squash chunks in the slow cooker and sprinkle with the brown sugar, 1/2 teaspoon salt, and 1/4 teaspoon white pepper. Pour the contents of the frying pan over the squash and add the broth. Cover and cook on the high-heat setting for 3 hours or the low-heat setting for 6 hours.
Finish the soup. Using a blender or food processor, working in batches, process the squash mixture to a smooth puree. Return the soup to the slow cooker to keep warn until serving. Season to taste with salt and pepper. Ladle into bowls, garnish with a dollop of the sour cream and chives (or parsley) and serve.