Place rack in the center of oven and preheat oven to 350°F. Coat 2, 8-inch round cake pans with cooking spray, then line the bottoms with parchment paper. Coat parchment paper with cooking spray and lightly dust with flour. Set aside.

Prepare the cake: In a large mixing bowl, beat the dark brown sugar and butter at medium speed until well blended, about 3 minutes. Beat in eggs, 1 at a time, until well incorporated. In a separate bowl, sift all purpose flour, whole wheat flour, baking powder, baking soda, and salt, whisking well. Add one-third of the flour mixture to sugar mixture, then 1/2 cup milk, then the next third of the flour mixture, remaining milk, and finally the remaining flour mixture. Beat gently after each addition until well combined.

Divide the batter between the prepared pans and sharply tap pans once on counter to remove any air pockets. Bake for 25 minutes or until a wooden pick inserted in center comes out clean, rotating the pans’ position once halfway through. Cool cake in pans for 10 minutes on a wire rack, then gently remove the cakes from pans. Remove wax paper. Allow cakes to cool completely on wire rack.

Prepare the frosting: Beat cream cheese and butter together on medium speed. Beat in melted white chocolate and vanilla. On low speed, slowly beat in the powdered sugar, until combined. Frost and assemble cake.