Soak the tapioca in warm water to cover for 1 hour. Drain and set aside.
Preheat the oven to 375 degrees. Lightly butter a 12-inch oval or gratin baking dish and place it in a large brownie pan.
Mix the tapioca, soy milk, sugar, and salt in a heavy nonaluminum saucepan. Bring the soy milk to a boil over medium heat while whisking. Lower the heat and let the mixture simmer 10 minutes, whisking occasionally. Remove from the heat and quickly whisk in the egg yolks, vanilla extract, cinnamon, and nutmeg.
In another bowl, beat the egg whites with the cream of tartar until just stiff. Whisk 1/2 a cupful into the tapioca, then fold the remaining in. Pour the tapioca mixture into the prepared baking dish.
Place the baking dish and brownie pan in the middle shelf of the oven and pour several inches of boiling water around the outside. Bake about 35 minutes, until the top is puffed and golden. The inside will still be a little soft. Remove, cool slightly, and serve warm or at room temperature.