Preheat the oven to 350 degrees F. Spray a 9 1/2 to 10-inch springform pan with nonstick spray. Line the bottom with a round of parchment paper and then spray again.
Layer the apple slices in the pan until they come about 2/3 of the way up the side. (I went a little higher than that and it worked out fine). Sprinkle the cinnamon-sugar over the apples.
Prepare the batter by beating the eggs and sugar until light and fluffy. Add the oil and the vanilla and beat well, then stir in the flour. Pour the batter on top of the apples, and sprinkle with additional cinnamon-sugar. Tap the pan on the counter a few times to allow the batter to sink down and around the apples.
Bake for 1 hour and 20 minutes, or until a skewer inserted into the center of the cake comes out clean. Cool *completely* in the pan. If you try to remove the cake from the pan while it is still warm, it will tend to break apart. I refrigerated my cake before slicing, and that worked out well. Serve slices with ice cream (warm individual slices in the microwave, if desired).
You want "caster sugar" for this recipe: It is a finer grind than table sugar, but not as fine as powdered. Look for "Baker's Sugar" at the store, or simply put regular white sugar in your food processor and give it a few whirls to create superfine sugar.