Preheat the oven to 400 degrees F.
Butter a 9-inch springform pan and set it aside.
In a large bowl, combine the flour, sugar, baking powder, and sea salt, and stir to blend. Add the vanilla extract, eggs, oil, and milk, and stir until well blended. Add the apples and stir to thoroughly coat them with the batter.
Spoon the mixture into the prepared cake pan. Place the pan in the center of the oven and bake until fairly firm and golden, about 25 minutes.
Meanwhile, prepare the topping: In a small bowl, combine the sugar, egg, and melted butter, and stir to blend. Set it aside.
Remove the cake from the oven and pour the topping mixture over it. Return the cake to the oven and bake until the top is a deep golden brown and the cake feels quite firm when pressed with a fingertip, about 10 minutes.
Transfer the cake pan to a rack and allow to cool for 10 minutes. Then run a knife around the sides of the pan, and release and remove the springform side, leaving the cake on the pan base. Serve at room temperature, cut into thin wedges.
Recipe courtesy of THE PARIS COOKBOOK (HarperCollins, 2001) by Patricia Wells