Preheat the oven to 325°F. Lightly grease a 9" x 13" pan. Or you can also prepare two 8" round pans and make a layer cake.
Mix ingredients together. Mix all of the ingredients except the apples and nuts in a large bowl. As soon as the mixture comes together and becomes pretty uniformly crumbly, stop mixing.
Add the apples and nuts. Mix until the apples release some of their juice and the stiff mixture becomes a thick, creamy batter, somewhere between cookie dough and brownie batter in consistency. Don't worry if the mixture doesn't immediately loosen up; this will take about 3 minutes at low speed in a stand mixer.
Bake the cake. Spread the batter in the prepared pan(s), smoothing it with your wet fingers. Bake the cake for 45 minutes for 9" x 13" pan or for 38 minutes for two 8" round pans. A toothpick or paring knife inserted into the center should come out clean, or with just a few wet crumbs clinging to it. The temperature at the center of the cake will be about 205°F. Remove the cake from the oven and place it on a rack to cool a bit while you make the frosting.
Make the frosting. Melt the butter in a medium-sized saucepan over medium heat. Stir in the brown sugar and salt and cook, stirring, until the sugar starts to melt and the mixture becomes fairly smooth. Add the milk and bring to a boil.Let the syrup cool in the bowl for 10 minutes.
Pour the confectioners' sugar into the warm syrup in the bowl, then add the vanilla extract. Whisk until everything is thoroughly combined. You need to work fast here; the frosting stiffens up quickly as it cools.
Pour the warm frosting onto the cake(s), spreading it over the entire surface.