Set your oven to 350°F and use the convection setting if you have it. Let it heat fully while you prepare everything else. A properly heated oven helps the cake bake evenly.
Mix the flour, softened butter, and oil into a paste. Brush it into every ridge and crevice of the pan. Set the pan aside while you mix the batter.
Beat the softened butter and brown sugar in a large bowl for two to three minutes. The mixture should turn light and fluffy. This step adds air, which gives the cake a soft crumb.
Add the eggs one at a time, mixing well after each one. Then stir in the vanilla and apple flavoring. The batter should look smooth and creamy.
In a separate bowl, whisk together the flour, cinnamon, baking powder, and pudding mix. Whisking removes lumps and spreads the leavening evenly. Keep this bowl close, because you will add it in stages.
Add the dry mixture to the wet mixture in alternating additions with the cider and sour cream. Begin and end with the dry ingredients. Mix only until everything is combined.
Spread the batter evenly into the prepared pan. Bake for 50 to 55 minutes, until a toothpick comes out clean. A few moist crumbs are fine.
Let the cake rest in the pan for 10 to 15 minutes. Then flip it onto a cooling rack. Let it cool completely before adding any toppings.
Stir the sugar, cinnamon, and salt together. While the cake is still slightly warm, brush it with the melted butter. Sprinkle the cinnamon sugar evenly over the top.
Heat the sugar and water in a saucepan over medium heat. Stir until the sugar dissolves, then stop stirring. Cook until the syrup turns deep amber, and watch it closely.
Take the pan off the heat and whisk in the cream and corn syrup carefully. It will bubble up fast. Return to low heat, stir until smooth, then add the vanilla and butter and let it cool.
Combine the white chocolate chips, cream, cream cheese, and butter in a microwave-safe bowl. Heat in 30-second bursts, stirring between each. Whisk in the powdered sugar if you want a thicker, sweeter icing.
Once the cake is fully cool, drizzle the caramel over the top. Pour or spread the icing next. Let it set for 10 to 15 minutes before slicing.