Preheat the oven to 350 degrees F. Fill a baking dish with ½-inch of water. Place the squash halves cut-side down. Place the carrots in the baking dish beside the squash. Cover with aluminum foil.
Bake the squash until it is soft and can be pierced easily with a fork, about 45 minutes.
In a saucepan, heat olive oil over medium heat. Add diced onion, minced garlic, and grated ginger. Saute for about 5 minutes until the onion starts to get translucent.
In a large pot, heat chicken (or vegetable) stock.
If you have an immersion blender, scoop out the flesh of each squash half and place it in the pot with the chicken stock. Add carrots, onion, garlic, and ginger to the pot. Use the immersion blender and blend the vegetables until smooth and creamy.
If you do not have an immersion blender, you can use a blender or a food processor. Take one cup of hot chicken stock and place it in the blender. Scoop out the squash from one half and place it in the blender. Place the carrots in and blend until smooth. Pour the smooth soup into the large pot.
Scoop out the squash from the other half and place it in a blender. Add the cooked onion, garlic, and ginger to the blender and blend until smooth. Pour soup into the pot. Add cinnamon, nutmeg, salt, and pepper to the soup. Stir well to combine. Serve the soup hot.
If desired, you may add one tablespoon of whipping cream or coconut milk to each serving for an even creamier soup.