Preheat an oven to 400 degrees F (200 degrees C). Line a baking sheet with aluminum foil.
Combine the brown sugar with 1/4 cup butter, cinnamon, and five spice powder in a microwave safe bowl. Cook in the microwave on High until the butter has completely melted. Place the acorn squash cut side up onto the aluminum foil; brush with the brown sugar mixture.
Bake in the preheated oven until the squash is tender and golden brown, about 1 1/2 hours. Remove from the oven, allow to cool, and scoop out and reserve the flesh.
Melt the remaining 1/4 cup of butter in a large pot over medium heat. Stir in the onion, celery, and bell pepper. Cook and stir until the onion has softened and turned translucent, about 5 minutes. Add the acorn squash, garlic, and seafood stock. Bring to a boil over high heat; reduce heat to medium-low and simmer 30 minutes.
Stir the cream and basil into the simmering soup; season with nutmeg, salt, and white pepper. Pour the soup into a blender, filling the pitcher no more than halfway full. Hold down the lid of the blender with a folded kitchen towel, and carefully start the blender, using a few quick pulses to get the soup moving before leaving it on to puree. Puree in batches until smooth and pour into a clean pot. Alternately, you can use a stick blender and puree the soup right in the cooking pot. Once pureed, add the crawfish tails, and simmer a few minutes until hot.