Grease a 9-by-5 inch loaf pan. Line it with a 9-inch tall parchment paper rectangle that is at least 12 inches wide. (The extra paper hanging over the pan's edges will make removing the cake easier.) Also, lightly grease the parchment.
In a medium or large bowl, beat the eggs and the sugar together until the mixture becomes light and fluffy, about 3 minutes.
Mix the hot melted butter with the cold milk to cool it down. Add the butter and milk mixture and baking powder to the beaten eggs and mix to combine. Note 2
Add the flour to the bowl and mix everything. A very loose or runny batter should form. Set the bowl aside.
Peel the apples, slice them into quarters, and remove the cores. Use a mandolin to make thin apple slices, or cut the apples very thinly by hand. Add the apple slices to the batter as you slice to prevent them from oxidizing and turning brown. Note 3 & 4
Mix the apple slices thoroughly with the batter to coat each one. Pour the mixture into the prepared pan.
(Optional) Reserve 40 to 50 of the prettier apple slices. Add the remaining apple and batter to the pan. Starting from one of the short ends, take an apple slice and slightly overlap it on top of the previous one to create a shingled pattern.
Use a spatula to gently press down on the cake to remove any air bubble in-between the slices and make the layers more even. You also gently tap the pan on the counter.
Bake the cake in a 375 F oven for 55-65 minutes, or until set.
Take the cake out of the oven and let it cool in the pan for at least 15-20 minutes before removing it. Loosen the edges of the cake from the pan. Pull the handles of the parchment paper to remove the cake.
You can serve the cake warm or cold. Optionally add a sprinkle of powdered sugar, whipped cream, or ice cream with the cake.