Preheat oven to 350 degrees F (177 degrees C) and place rack in center of oven. Butter (or spray with Pam) a 9 inch (23 cm) glass pie plate.
In a small bowl whisk together one egg and 1 tablespoon from the 1/2 cup milk. Set this aside as it will be used as a glaze for the top of the cake.
In a large bowl, whisk together the flour, sugar, baking powder, and salt. Cut the butter into small pieces and blend into the flour mixture with a pastry blender or two knives. The mixture should look like coarse crumbs. In a separate small bowl whisk together the remaining one egg, remaining milk, and vanilla extract and add to the flour mixture, stirring just until the dough comes together. Do not overmix the dough.
Transfer to a lightly floured surface and knead dough gently four or five times and then divide the dough in half. Roll or pat one half of the dough into a 9 inch (23 cm) circle and transfer to the pie plate and pat onto the bottom and up the sides of the pie plate.
In a separate bowl toss together the cut apples, sugar and cinnamon. Spread the apples evenly over the bottom of the dough in the pie plate.
Roll the remaining dough into a 9 inch (23 cm) circle, on a lightly floured surface, and gently place the dough over the apples. With your fingers seal the edges of the top and bottom crusts. Brush the top of the dough with the egg and milk glaze, and sprinkle with white sugar. Cut a slit in the center of the dough to allow the steam to escape.
Bake in the preheated oven for about 35 - 45 minutes or until the pastry is nicely browned and a toothpick inserted into the center of the cake comes out clean.
Cool on a wire rack. Serve warm with a dollop of whipped cream or ice cream.