In a large bowl, stir together the peeled and diced apples, walnuts and the sugar; let sit for 30 minutes (this will bring some of the natural juices out of the apples).
Preheat oven to 350°F and grease a 9x13" pan; set aside.
Pour in the vegetable oil, eggs and vanilla; stir well.
Add the flour, baking soda, cinnamon and salt; stir to combine.
Pour the cake batter into prepared pan and bake for 45-55 minutes, or until a toothpick poked into the center of the cake comes out clean (or with a few moist crumbs).
For the Icing
In a saucepan, combine the butter, brown sugar and milk.
Place on the stove-top on medium heat and cook until the brown sugar is dissolved.
Remove from the heat and stir in the powdered sugar and vanilla.
Pour the warm icing over the cake (cake doesn't need to be completely cooled).