Preheat the oven to 350(F). Shred carrots and transfer them to a bowl lined with paper towels. Let them sit while you make the batter.
Combine all the dry ingredients (flour, cinnamon, baking powder, baking soda, salt) in a bowl and mix to combine.
In another bowl, combine the brown sugar, vegetable oil, and applesauce. Whisk until smooth. Add eggs and vanilla and mix again.
Slowly add the dry ingredients into the wet ingredients and mix until almost combined. Squeeze any remaining juice from the shredded carrots and add them to the batter. Fold the batter until the carrots are just combined.
Grease a baking sheet (18"x13") and pour the batter in. Spread the batter out so it is equally distributed in the pan. Bake at 350 for 20-23 minutes. A toothpick inserted in the middle should come out clean.
While the cake is cooling, make the frosting. Add the butter and cream cheese to a bowl. Using a stand or hand mixer, beat the butter and cream cheese for 2 minutes, until lighter and fluffy. Add the vanilla and powdered sugar and beat an additional 3 minutes. The frosting should be well combined and smooth.
Using a lid or small bowl (something round), cut out small circles of cake. Place the lid on top of the cake, then cut around it with a sharp knife. Repeat with the entire cake. You should have about 24 small rounds of cake.
Pipe frosting onto one small cake. Top it with another small cake and pipe more frosting on the top. Continue with all the cakes. If desired, crumble extra cake on top of the prepared cakes and top with edible spring flowers! Enjoy!