1. Make cake: Preheat oven to 350°F. Mist a 9-by-13-inch baking pan with cooking spray. Spread nuts in a single layer on a baking sheet; bake until lightly toasted, 5 to 7 minutes, stirring halfway through. In a bowl, mix flour, baking soda, cinnamon, ginger and salt.
2. In a large bowl, whisk butter, sugar, eggs and vanilla. Stir flour mixture into butter mixture. Fold in apples and walnuts. (Batter will be very thick.) Spread batter in pan. Bake until a toothpick inserted in center comes out clean, 45 to 50 minutes. Place on a wire rack and let cake cool completely.
3. Make frosting: In a mixing bowl, using an electric mixer on medium-high speed, beat cream cheese and butter. Beat in lemon juice. Reduce speed to low, then add confectioners' sugar gradually. Fold in zest. Spread frosting on cake; sprinkle with ginger.