Preheat oven to 325 degrees. Place sliced apples in a 9-inch spring form pan and toss with 1 tablespoon water. Cover tightly with tin foil and bake for 30 minutes, until apples are tender.
Meanwhile, whisk together 1 cup flour, 1 cup sugar, baking powder, and salt in a medium bowl. In a large liquid measuring cup, whisk together oil, milk, 1 whole egg, and vanilla. Pour the liquid into the dry ingredients and fold together until incorporated. Remove one cup of batter and set aside. Add the egg yolks to the original bowl of batter and whisk to combine.
Remove apples from the oven and toss with the calvados and lemon juice. Allow to cool for about 10 minutes. Drain the apples to remove most of the the liquid. Add the apples to the original bowl of batter and stir to combine.
Line the spring form pan with a parchment paper round. Pour the batter with the apples into the pan.
To the reserved one cup of batter, stir in 2 tablespoons of flour. Pour on top of the apple/batter mixture and smooth the top with an offset spatula. Sprinkle remaining 1 tablespoon of sugar evenly on top.
Bake for 1 hour to 1 hour 15 minutes. The top should be golden brown and a toothpick inserted into the center of the cake should out clean. Allow the cake to cool for 30 minutes before removing the rim of the springform pan. This cake is delicious served slightly warm, perhaps with a dollop of lightly sweetened whipped cream, but it can be kept at room temperature for several days.*Adapted from America's Test Kitchen