Start by preheating your oven to 350°F (175°C). Lightly grease a 9×13-inch baking pan or line it with parchment paper to prevent sticking.
Wash and peel the carrots, then finely grate them using a hand grater or food processor. Set them aside for later.
In a large mixing bowl, combine the flour, baking soda, baking powder, salt, cinnamon, nutmeg, and cloves. Whisk them together until evenly blended.
In a separate bowl, whisk together the eggs, brown sugar, granulated sugar, oil, and vanilla extract. Stir until the mixture is smooth and fully combined.
Slowly add the dry ingredients to the wet mixture, stirring gently as you go. Once everything is well mixed, fold in the grated carrots until they’re evenly distributed throughout the batter.
Pour the batter into your prepared pan and smooth out the top. Bake for 25 to 30 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool completely in the pan on a wire rack. While it cools, prepare the frosting: beat softened butter and cream cheese together until smooth. Add vanilla extract, a pinch of salt and powdered sugar, then beat until the frosting is light and fluffy.
Once the cake has cooled, spread the frosting evenly over the top. Cut into bars and serve. For neater slices, you can refrigerate the cake for 15 to 20 minutes before cutting.