Butter a 9-inch springform pan and set it aside.

In a large bowl, combine the flour, sugar, baking powder, and 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 oven and bake until fairly firm and golden, about 40 minutes.

Meanwhile, prepare the topping. In a small bowl, combine the sugars, eggs 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 15 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 the cake warm with the chilled pumpkin crème anglaise.

To make the creme anglaise, pour the milk into a heavy saucepan. Scrape the vanilla bean seeds into the milk, then add the vanilla bean. Heat the saucepan over medium heat.

In a separate bowl, whisk together the egg yolks and sugar.

Reduce the heat of the milk mixture to low and slowly whisk ½ cup of the egg mixture into the hot milk. Then whisk in the remaining egg mixture into the hot milk. Cook, stirring constantly, until the mixture is thick enough to coat the back of a wooden spoon, about 15 minutes. Whisk in the pumpkin. Cook another 5 minutes while stirring constantly.

Strain the crème anglaise through a fine sieve into a bowl. Cool to room temperature, then cover and refrigerate at least three hours. Strain again. Serve the crème anglaise chilled.