To Make the pastry cream Filling: In a sauce pan, combine the milk and 1/3 cup of sugar. Bring to a boil, stirring to dissolve sugar. Remove from heat. In a small bowl, combine 1/3 cup of sugar and 2 tablespoons flour. Beat in the eggs and vanilla.
Mix 1/2 cup of hot milk into egg mixture, then pour the egg mixture back into the saucepan with the hot milk. Return to heat and bring to a boil. Continue cooking over medium heat until mixture thickens and becomes smooth. Remove from heat and let cool for 1 hour.
Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9 inch springform pan. Combine and sift the flour and baking powder. Set aside
Cream 1 1/8 cup sugar and 3 eggs until light and fluffy. Fold in the sifted flour mixture in three increments, being careful not to overmix.
Put half of the dough into the greased pan. Spread the dough so that it covers the bottom of the pan. Place pastry cream to within 3/4 inch of the edge. Add the second half of the cake dough, making sure to enclose all of the filling.
Bake at 350 degrees F (175 degrees C) for 30 to 40 minutes or until golden brown.