Preheat the oven to 325°F. Grease and flour a 9" x 13" pan.

To make the cake: Mix all of the ingredients except the apples and nuts in a large bowl. Beat until well combined; the mixture will be very stiff, and may even be crumbly.

Add the apples and nuts, and mix until the apples release some of their juice and the stiff mixture becomes a thick batter, somewhere between cookie dough and brownie batter in consistency. Spread the batter in the prepared pan, smoothing it with your wet fingers.

Bake the cake for 45 minutes, or until a cake tester or toothpick inserted into the center comes out clean, or with just a few wet crumbs clinging to it.

Remove the cake from the oven and place it on a rack to cool completely; don't remove the cake from the pan.

To make the frosting: Melt the butter in a small pan over medium heat. Stir in the brown sugar and salt and cook, stirring, until the sugar melts.

Add the milk, bring to a boil, and pour into a mixing bowl to cool for 10 minutes.

After 10 minutes, stir in the confectioners' sugar and vanilla. Beat well; if the mixture appears too thin, add more confectioners' sugar. Spread on the cake while frosting is still warm.