Do not peel the apples. Cut them into quarters, core them, and cut them into 1/4-inch (6-mm) thick slices.
Meanwhile, in a 10-inch (25-cm) cast-iron skillet, melt the butter over medium heat.
Continue heating for 4 to 5 minutes more, gently tipping the pan back and forth, until the butter smells toasty and has turned a medium shade of amber (it’s hard to judge color against the jet-black metal, so spoon a bit out onto a white plate to get a better look).
Remove the pan from the heat and stir in the maple syrup.
Arrange the apple slices on top in concentric circles, tightly overlapping the slices to fit as many as possible. Set aside.
Preheat the oven to 350°F (175°C).
Cream together the butter and sugar with an electric mixer.
Mix in the eggs one at a time.
With the mixer running, add the milk slowly and mix well.
In a separate medium bowl, sift together the flours, baking powder, salt, cinnamon, and nutmeg.
Add half of the dry ingredients to the batter and mix until just combined.
Then mix in the apple cider.
Add the rest of the dry ingredients and mix just until incorporated.
Pour the batter on top of the apples and smooth it evenly to the edges.
Bake the cake for 35 to 40 minutes or until a toothpick inserted into the center comes out clean and the edges are lightly browned and pulling away from the sides of the pan (if you’re using a cake pan instead of a skillet, you may need to increase the baking time by 5 to 10 minutes).
Remove the pan from the oven and let it cool until it's no longer hot to the touch. If you’re strong, invert the cake onto a serving plate by setting the plate over the top of the skillet, holding them firmly together, and flipping. Or bring the skillet to the table and slice the cake, inverting each piece onto a plate, apple-side up. Serve warm.