Cut squash in half and remove all the seeds and inner fibers. (It's best to remove with a grapefruit knife) Place skin side in a baking dish with a little water and roast in a preheated 350 degree oven for 30 minutes. Remove all skin from squash when cool. Cut squash into two inch square.
Place squash with fresh garlic and fresh ginger in a large pot with 1/2 cup of chicken stock and sweat over low heat for 10 minutes. Add remaining stock and bring to a boil. Turn down to light simmer and cook until squash is tender approximately 30 minutes. Add remaining ingredients except for shiitake mushrooms. Puree in a food processor until smooth. Remove the stems from the Shiitake mushrooms and slice a 1/4-inch thick. Simmer in squash soup for 2 minutes.
Ladle six ounces into each serving bowl and garnish with slice basil.