Preaheat oven to 325 degrees Fahrenheit. Prepare the springform by spraying a 9-inch springform pan with vegetable oil spray. In a large microwave-proof bowl, add apples, cover with plastic wrap, and microwave three minutes on high heat. Carefully remove the plastic wrap, add lemon juice and rum to the bowl and toss with the apples.
In a medium bowl, add one cup of flour, one cup of sugar, baking powder, and salt and mix well. In a large bowl, add oil, milk, one egg, and extract and whisk until combined. Add flour mixture to the milk mixture and mix until combined.
Remove one cup of the cake mixture to a medium bowl. Add the remaining three tablespoons of flour and whisk well. This will be the top cake layer. Set aside.
Add three egg yolks to the original cake batter and mix to combine. Stire apples into the cake batter. Pour the batter into the prepared pan. Gently shake any bubbles out of the batter.
Pour top cake batter over batter in pan. Smooth the surface to get rid of any bubbles. Sprinkle the remaining two tablespoons of sugar evenly over the top of the batter. Bake for about 1 1/2 hours, or until golden brown and a toothpick inserted in the cake comes out clean. Remove pan to a wire cooling rack and allow to completely cool. Loosen the sides of the cake by running a knife around the edges of the pan. Remove sides of pan and sprinkle liberally with powdered sugar. Serve with homemade whipped cream.