Very lightly adapted from Dorie Greenspan's Baking Chez Moi. The custardy squares are essentially an autumnal clafoutis; they're lovely. I've added the option below of not using a mandoline to slice the apples; I made it without, and without being too persnickety, and it was delicious nonetheless. —Nicholas Day
unsalted butter, melted and cooled (but still liquid)
Confectioners' sugar (optional)
Heat the oven to 400° F. Butter an 8-inch square baking pan and line the bottom with parchment paper.
Peel the apples. If you have a mandoline, slice the apples thinly, turning when you reach the core. (The slices should be thin but not transparent.) If you don't have a mandoline, simply core and slice as thinly as you manage. (Don't worry about the slices being impossibly precise or thin.)
In a bowl, whisk together the flour and baking powder.
In a large bowl, whisk together the eggs, sugar, and salt for a couple of minutes, or until the sugar dissolves and the eggs become pale. Whisk in the vanilla, then the milk and the melted butter. Add the flour and whisk until smooth. With a spatula, gently fold in the apples until each slice is coated. Scrape the batter into the pan and roughly even out the top.
Bake the cake for 40 to 50 minutes or until golden and uniformly puffed. A skewer in the middle will come out clean. Transfer to a rack to cool, then slice and dust with the optional confectioners' sugar.