Preheat oven to 400F
Butter a 9 inch springform pan very very well - this cake tends to stick a lot
In a large bowl, combine flour, 1/3 cup sugar, baking powder, and salt. Stir to blend
Add vanilla, 2 eggs, oil, and milk. Stir until well-blended
Add the apples and stir to thoroughly coat in batter. Spoon mixture into prepared pan, place in center of oven, and bake for 25 minutes until golden brown and fairly firm
While it's cooking, combine 1 egg, 1/3 cup sugar, and butter to form a topping. Pour the topping over the cake and return to oven for 10 minutes, until the top is a deep golden brown and it feels quite firm when pressed with a fingertip
Transfer cake pan to rack and allow to cool for 10 minutes. Run a knife around the sides of the pan and release and remove from springform, leaving on base