Preheat the oven to 400 degrees. Halve the squash lengthwise with a sharp knife and scoop out the seeds and fibers. Place cut side up in a shallow baking dish and cover tightly with foil. Bake for 45 to 50 minutes, or until tender. Set aside until cool enough to handle.
Heat the oil in a soup pot. Add the onion and sauté over medium-low heat until golden, about 8 to 10 minutes. Add the apple dice, water, bouillon cube, and spices. Bring to a simmer, and then simmer gently, covered, until the apples are soft, about 10 minutes.
In a food processor, puree the squash with 1/2 cup of the milk or soymilk until completely smooth. Transfer to a bowl.
Transfer the apple-onion mixture to the food processor and puree until completely smooth. Transfer both this and the squash puree back into the soup pot and stir together. Add the remaining milk or soymilk; using a bit more if the puree is too thick.
Bring the soup to a simmer once again, and then cook over low heat for another 5 to 10 minutes, or until well heated through. Season to taste with salt and white pepper. Serve at once, or let the soup stand off the heat for an hour or two, then heat through as needed before serving.