First, prepare the apples. Peel and core them, then cut them into quarters. Slice the quarters lengthwise into thin wedges. One apple should yield 16 of these slices. Put them on a pie plate, cover them with a sheet of parchment paper or waxed paper, and microwave for about two and a half minutes, until they are just barely translucent. Put them in a medium bowl, add the lemon juice and Calvados, and toss everything together with your hands. Then set the bowl aside to let the apple slices cool.
Prepare the springform pan with cooking oil. When I made this, I also encased the bottom of the pan with aluminum foil to stave off leaks. Set the pan on a baking sheet and preheat the oven to 325ºF.
Next, put 1 cup of flour, 1 cup of sugar, the nutmeg, salt and baking powder into a medium bowl and gently stir together with a fork.
In a large bowl, whisk together the whole egg, the milk, oil and vanilla. when everything is smooth, add in the dry ingredients and whisk again, just until everything is combined. Measure out 1 cup of this batter and put it in a small bowl. Set that aside for now.
Add the one egg yolk to the large bowl and stir it in. Then fold in the apples—they should be mostly cooled when you do this. Scrape this batter into the springform pan and even out the top. You need the top to be uniformly smooth.
Add the extra 2 tablespoons of flour to the small bowl and mix together until it is uniform. Then carefully spread this enhanced batter on top of the batter in the springform pan. Try to make it smooth and uniform in thickness.
Finally, sprinkle the last 2 tablespoons of sugar all over everything. Put the cake on the middle rack in the oven.
Bake until the center of the cake is set, the top is golden, and a tester inserted in the center comes out clean. This will take about 70 minutes or so.
Transfer the pan to a rack and cool for 10 minutes. Then slip a thin knife around the sides of the pan, loosen and remove the ring, and let the cake cool completely. Serve this, if you like, with whipped cream or a little vanilla ice cream, or, as my mother used to say, just a hungry mouth.