Preheat the oven to 325°F. Grease and flour a 9 x 13-inch pan.
Mix all of the ingredients except the apples and walnuts in a large bowl. Beat until well combined; the mixture will be crumbly. Add the apples and walnuts, and mix until the apples release some of their juice and the crumbly mixture becomes a thick batte
Spread the batter in the prepared pan, smoothing it with your wet fingers. Bake the cake for 45 minutes, or until the cake tests done. Remove from the oven and place on a rack, to cool completely; don't remove the cake from pan. When the cake is completel
Melt the butter in a small pan over medium heat. Stir in the brown sugar 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.