Step 1

Preheat oven to 350 degrees F (175 degrees C).

Step 2

Whisk all-purpose flour, whole wheat flour, baking soda, baking powder, salt, and cinnamon together in a bowl. Whisk oil, white sugar, 1/2 cup brown sugar, eggs, and vanilla together in a separate bowl. Stir flour mixture into sugar mixture; fold apples and 1 1/2 cups walnuts into batter. Pour batter into a 10x13-inch baking dish.

Step 3

Bake in the preheated oven until cake is set, about 1 hour. Pierce surface of cake in several places with a large fork.

Step 4

Stir brown sugar, butter, and milk together in a large microwave-safe bowl. Microwave on high until sugar is dissolved and mixture boils, about 2 1/2 minutes. Sprinkle 1/2 cup walnuts over hot cake and pour brown sugar mixture over the top.