Center the oven rack, preheat oven to 375°, and line a lightly-greased 8x8 cake pan with parchment paper (as illustrated below). Peel, core, and slice the apples 1/8- to 1/4-inch thick. Put the apple slices in a large bowl, and toss first with the lemon juice, and then with the cinnamon-sugar mixture. Set aside.
In a medium bowl, whisk together the flour, baking powder, salt, and nutmeg. Set aside.
In the bowl of a standing mixer (or a large bowl if using electric beaters), beat the softened butter at medium speed for 1 minute. Then add the sugar, and beat at medium speed until the mixture is light and fluffy. With the machine running, gradually add the eggs, followed by the oil and vanilla. At low speed, gradually add the flour mixture, beating only until incorporated.
Pour the batter into the prepared cake pan, and then smooth the top with a spatula. Arrange the apples on top, in 3 slightly overlapping rows (as illustrated below).
Bake until a skewer inserted into the batter part of the cake comes out clean -- about 55 minutes. Unmold the cake onto a board or platter, and cut into 9 equal squares. Serve warm or cold, with or without vanilla ice cream.