01 - Preheat oven to 350°F. Line a 9×13-inch baking pan with parchment paper, leaving overhang on the sides for easy removal.
02 - In a medium bowl, whisk together the all-purpose flour, whole wheat flour, baking powder, baking soda, salt, ground ginger, cinnamon, nutmeg, and cloves until evenly distributed.
03 - In a large bowl, beat the brown sugar, vegetable oil, eggs, vanilla extract, and applesauce until smooth and well combined.
04 - Gently fold the grated carrots, squeezed zucchini, and crystallized ginger (if using) into the wet mixture.
05 - Add the dry ingredient mixture to the wet ingredients and stir until just combined. Be careful not to overmix.
06 - Spread the batter evenly into the prepared baking pan. Bake for 30 to 35 minutes, or until a toothpick inserted in the center comes out clean. Cool completely in the pan on a wire rack.
07 - Beat the softened cream cheese and butter together until smooth. Mix in the sifted powdered sugar, fresh lemon juice, lemon zest, and vanilla extract until light, fluffy, and well combined.
08 - Once the bars are fully cooled, spread the lemon cream cheese frosting evenly over the top. Cut into 12 bars and serve. Store leftovers refrigerated in an airtight container.