Prepare the Oven and Pan: Preheat your oven to 350°F. Grease a 13 × 18-inch rimmed baking sheet with butter and line it with parchment paper, leaving about a 1-inch overhang on the long sides. Lightly grease the parchment paper as well.
Mix the Cake Batter: In a stand mixer equipped with a paddle attachment, combine the canola oil, buttermilk, eggs, granulated sugar, brown sugar, and vanilla extract. Beat the mixture on medium speed until it becomes smooth. Add the grated carrots and shredded coconut, stirring until well mixed. Reduce the mixer speed to low and gradually incorporate the flour, baking powder, baking soda, cinnamon, and salt. Blend the ingredients until there are no visible lumps of flour, which should take around 2 minutes.
Bake the Cake: Pour the batter evenly into the prepared baking sheet. Smooth out the top, then bake for 25 to 30 minutes, or until the center is set and does not jiggle. Once baked, remove the cake from the oven and carefully run a knife along the edges to loosen it. Allow the cake to cool completely in the pan, which will take about an hour.
Prepare the Frosting: In a stand mixer, beat together the cream cheese and softened butter until the mixture becomes light and airy, about 2 minutes. Gradually add the confectioners' sugar and vanilla extract, continuing to beat until smooth and fluffy, approximately 2 more minutes. If you'd like a fruity touch, mix in the freeze-dried berry powder until fully incorporated.
Assemble the Cake Layers: Using the parchment paper to lift the cake, carefully remove it from the pan. With a serrated knife, slice the cake evenly into three rectangular layers, each about 13 × 6 inches in size. Place the first layer on a serving platter and spread one-third of the frosting evenly across the surface, making sure to cover all edges. Add the second layer on top, frost it in the same way, and repeat with the final layer. Use the remaining frosting to coat the top of the cake.
Serve or Store the Cake: Once the cake is fully assembled, you can serve it immediately. If you prefer to store it, place it in an airtight container and refrigerate for up to one day for best results.