Position a rack in the middle of the oven and preheat to 400 degrees. Grease an 8-inch square baking pan with a little butter.

Slice the apples from top to bottom using a mandoline, Benriner or sharp knife, turning the fruit 90 degrees each time you reach the core. The slices should be about 1/16 inch thick -- elegantly thin, but not so thin that they’re transparent and fragile. (If they’re a little thicker, that’ll be fine, too.) Discard or compost the cores.

In a small bowl, whisk together the flour and baking powder until combined.

In a large bowl, whisk together the eggs, sugar and salt until the sugar has just about dissolved and, more importantly, until the eggs are pale, about 2 minutes. Whisk in the vanilla extract, then the milk and the melted, cooled butter. Add the flour mixture and whisk to form a smooth batter.

Add the apples to the bowl, switch to a flexible spatula and gently fold in the apples, turning the mixture until each thin slice is coated in the batter. Scrape the mixture into the pan, smoothing 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 and puffed — make sure the middle of the cake has risen — and a knife inserted into the center comes out clean. Transfer to a wire rack and let cool for at least 15 minutes or completely.

Carefully cut into 9 equal squares in the pan (being careful not to damage the pan); or unmold the slab onto a rack, flip it onto a plate so it is right side up and then cut it into 9 squares. Either way, give the squares a dusting of confectioners’ sugar before serving, if you’d like.