To make the cream filling: In a saucepan, 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 180 degrees C. Grease and flour a 23cm 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. Add a layer of pastry cream to within 2cm of the edge. Add the second half of the cake dough, making sure to enclose all of the filling.
Bake for 30 to 40 minutes or until golden brown. Dust with icing sugar.