Cut squash in half lengthwise and remove seeds and fibers. Place squash cut side down in a baking dish. Pour the stock into the dish. Place in the oven and bake for 20 minutes.
Scoop out the fruit from the apples, leaving a 1/4 border. Remove squash from oven, place apples cut side down in baking dish, fitting them around the squash. Return to oven and bake until squash and apples are just tender, about 15 to20 minutes more. Turn squash and apples over. Dot with butter and brown sugar. Return to oven for 10 minutes.
Remove squash and apples from oven. Transfer apples to a warm platter and reserve, covered. Scoop squash flesh into a medium bowl. Mash with the cream, walnut oil, cinnamon and ginger, adding salt and pepper, to taste. Scoop the squash into the apple halves and serve.