For the apples:

1. Peel and core the apples and dice them into small pieces.
2. Place the chopped apples in a sauté with brown sugar and salt. Over medium heat, cook the apples, stirring occasionally, until the sugar dissolves. Lower to medium-low heat and cook until the apples have softened. This should take 7-9 minutes. The sugar and juices from the apples will provide a syrupy mixture. Remove from heat and let the apple mixture cool to room temperature.

For the cake:

1. Preheat the oven to 350 degrees
2. Coat the bottom of the skillet and the sides generously with butter. You could use cooking spray, but I think butter is the better choice for a cast-iron skillet.
3. Add the butter and brown sugar to the bowl of an electric mixer with the paddle attachment and cream the butter and sugar, on high, for about 5 minutes until combined and a little bit fluffy.
4. Add the eggs, one at a time, and beat until combined
5. Scrape down the sides and bottom of the bowl and beat in the vanilla.
6. Whisk together some of the dry ingredients (flour, nutmeg, cinnamon, salt, and baking powder) in a medium bowl until combined.
7. Add the flour mixture to the bowl of the mixer and beat until just combined.
8. Remove the mixing bowl from the stand and using a rubber spatula, gently pour and fold the syrupy apples and all the juices into the cake batter. Mix until just combined.
9. Carefully pour batter into the prepared skillet and smooth it out evenly in the cast iron pan. 
10. Bake in a 350 degree oven until the top of the cake is evenly golden brown and the cake has a moist center. You’ll know it’s done when you insert a toothpick into the center of the cake and it comes out clean or with a few moist crumbs clinging to it. The baking time is approximately 1 hour. Keep a close eye on the cake, and if the edges begin to get dark brown, cover the cake loosely with aluminum foil. 
11. Let the cake cool completely before adding the frosting.

For the frosting:

1. Heat the butter in a small saucepan over medium-low heat and brown it. Stir regularly until it turns light brown and has a very nutty-like smell. This should take approximately 7 to 10 minutes. 
2. Once the butter is browned, whisk in the confectioner’s sugar, milk, and vanilla extract and mix until smooth with no lumps.
3. With an offset spatula, immediately spread the frosting over the top of the cake until coated. 
4. You’ll need to work fast because the frosting hardens quickly.
5. Cut the cake into pie-like slices and serve.