Make cake: Place a rack in center of oven and preheat to 350°F. Grease and flour 2 8-inch round pans. Whisk together flour through salt. In a bowl, mix apples and sugar.
Stir dry ingredients into apple mixture. Mix in eggs and oil. Divide batter between pans; bake until a toothpick inserted in center comes out clean, 25 to 30 minutes. Let cool for 10 minutes; then invert onto a rack to cool completely.
Make frosting: Beat cream cheese and butter until fluffy, 2 minutes. Reduce speed; add confectioners' sugar and vanilla. Mix until combined, 2 minutes more.
Place a cake layer on a platter; spread top with 1 cup frosting. Top with second cake layer. Frost top and sides and decorate with walnuts.