Adjust an oven rack to the middle position. Preheat the oven to 350 degrees F. Spray the bottom and sides of a 9-inch round cake pan with cooking spray. Line the bottom with a round piece of parchment and spray with cooking spray again.
Combine the all-purpose flour, whole wheat flour, baking powder, cinnamon and 1/2 teaspoon salt together in a medium bowl and whisk to blend all the ingredients.
Combine the sugar and eggs in the bowl of a stand mixer fitted with a paddle attachment and beat together until light yellow, about 1 minute. Add the apple sauce, butter and vanilla and beat until combined. Add the flour in two additions and beat until incorporated. Fold in the apples and pour the batter into the prepared cake pan. Bake until a toothpick inserted into the center of the cake comes out clean, 45 to 55 minutes. Cool completely in the pan on a rack. Invert onto a serving platter. Slice and serve.