Adjust oven rack to lower-middle position and preheat to 350°F (180°C). Lightly grease three 8-inch anodized aluminum cake pans and line with parchment (explanation and tutorial here). If you don’t have three pans, it’s okay to bake the cakes in stages; the batter will keep at room temperature until needed.
Combine white sugar, brown sugar, cinnamon, ginger, baking powder, salt, baking soda, nutmeg, cloves, vanilla, and eggs in the bowl of a stand mixer fitted with the whisk attachment. Mix on low to moisten, then increase to medium and whip until thick and fluffy, about 8 minutes. Meanwhile, whisk together all-purpose and whole wheat flours. Drizzle in the brown butter in a steady stream, then reduce speed to low and add the flours. Once smooth, fold in shredded carrots with a flexible spatula.
Divide batter between the prepared cake pans, about 14.5 ounces each. If you don't have three pans, the remaining batter can be held at room temperature up to 3 hours. Bake until cakes are golden, about 15-20 minutes. A toothpick inserted into the center will have a few crumbs still attached, and your fingertip will leave a slight indentation in the puffy crust.
Cool cakes directly in their pans for 1 hour, then run a butter knife around the edges to loosen. Invert onto a wire rack, peel off the parchment, and return cakes right side up (covered in plastic, the cakes can be left at room temperature for a few hours). Prepare the whipped cream frosting.