Preheat oven to 350 degree F. Place the 1/3 cup butter in a 9x9x2-inch baking pan. Place in oven about 5 minutes or until butter is melted. Halve apples; remove stems. Using a melon baller or small spoon, scoop out apple cores.
Sprinkle brown sugar over melted butter; stir to combine. Spread evenly over bottom of pan. Arrange nine of the apple halves, cut sides down, in butter mixture. Bake for 10 to 15 minutes or until butter mixture is bubbly.
Meanwhile, peel and coarsely shred the remaining three apple halves; set aside. In a medium bowl, stir together flour, granulated sugar, baking powder, ginger, and cinnamon. Add shredded apple, milk, the 1/4 cup softened butter, the egg, and vanilla. Beat with an electric mixer on low speed until combined. Beat on medium speed for 1 minute. Gently spoon batter over apple halves in pan, spreading evenly (some apple may still be exposed and some butter mixture may come to the surface).
Bake about 35 minutes or until a toothpick inserted near the center comes out clean. Cool cake in pan on a wire rack for 5 minutes.
Using a sharp knife or narrow metal spatula, loosen edges of cake from sides of baking pan. Carefully invert onto a serving platter. Spoon any butter mixture remaining in pan onto cake. Cool for 20 minutes more. Serve warm. If desired, serve cake with ice cream. Makes 9 servings.