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1
Center a rack in the oven and preheat the oven to 400 degrees F. Butter an 8-inch square baking pan.
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2
Slice the apples from top to bottom using a mandoline, Benriner or a sharp knife, turning the fruit as you reach the core.
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3
The slices should be about 1/16 inch thick-elegantly thin, but not so thin that they're transparent and fragile.
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4
Discard the cores.
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5
Whisk the flour and baking powder together in a small bowl.
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6
Working in a large bowl with a whisk, beat the eggs, sugar and salt together for about 2 minutes, until the sugar just about dissolves and, more important, the eggs are pale.
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7
Whisk in the vanilla, followed by the milk and melted butter.
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8
Turn the flour into the bowl and stir with the whisk until the batter is smooth.
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9
Add the apples to the bowl, switch to a flexible spatula and gently fold the apples into the batter, turning everything around until each thin slice is coated in batter.
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10
Scrape the batter into the pan and smooth the top as evenly as you can-it will be bumpy; that's its nature.
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11
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.
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12
Transfer the cake to a rack and allow to cool for at least 15 minutes.
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13
Using a long chefs knife, cut the cake into 8 squares 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.
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14
Either way, give the squares a dusting of confectioners' sugar before serving, if you'd like.
