Dorie mentions that you could add a splash of rum or Calvados if you like, or the zest of an orange or a lemon. Also, she mentions mixing up the fruit and trying it out with pears or quince, too.
Ever so slightly adapted from Dorie Greenspan
Center a rack in the oven and preheat the oven to 400 degrees F. Butter an 8-inch square baking pan and line the bottom with parchment paper.
Slice the apples using a mandoline, Benriner or a sharp knife, turning the fruit as you reach the core. The slices should be about 1/16th inch thick– elegantly thin, but not so thin that they’re transparent and fragile. Discard the cores.
Whisk the flour and baking powder together in a small bowl.
Working in a large bowl with a whisk, beat the eggs, sugar, salt and cinnamon together for about 2 minutes, until the sugar just about dissolves and, more important, the eggs are pale. Whisk in the vanilla, followed by the milk and melted butter. Turn the flour into the bowl and stir with the whisk until the batter is smooth. Add the apples, switch to a flexible spatula and gently fold the apples into the batter, turning everything around until each thin slice is coated in batter. Scrape the batter into the pan and smooth the top as evenly as you can–it will be bumpy; that’s its nature.
Bake for 40 to 50 minutes, or until golden brown, uniformly puffed– make sure the middle of the cake has risen–and a knife inserted into the center comes out clean. Transfer the pan to a cooling rack and allow to cool for at least 15 minutes.
Using a long knife, cut the cake into 8 squares (or as many rectangles as you’d like) in the pan (being careful not to damage the pan), or unmold the cake onto a rack, flip it onto a plate and cut into squares. Either way, give the squares a dusting of confectioners’ sugar before serving, if you’d like.
The cake is good at room temperature the day it’s made, or can also be refrigerated, covered, for up to 2 days and served chilled.