Preheat oven to 350°F Butter an 8-inch square glass baking dish.
SAUCE: Transfer 1 tablespoons milk to a small bowl. Heat remaining milk and sugar for 2-3 minutes or until steaming in a glass measuring cup in a microwave on medium(50%) power or in saucepan over medium heat on stove-top. Add cornstarch and vanilla to reserved milk; drizzle into hot milk mixture, stirring until sugar is dissolved. Set aside.
CAKE: In a large bowl, combine flour, sugar, baking powder and salt. In another bowl, whisk together egg, milk, butter and vanilla. Pour over dry ingedients and stir just until moistened. Spread into prepared baking dish. Gently pour warm milk mixture over cake batter without stirring.
Bake for about 30 minutes or until tester inserted in center of cake layer comes out clean. Let cool at least 15 minutes. Serve hot or warm. Dust with icing sugar (if using).
Scoop cake into serving bowl and spoon pudding sauce over top.